Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 7-Day Trial for You or Your Team.

Learn More →

Development of a Predictive Model for the Thermal Expansion Coefficient of Elastic Disordered Microporous Metal Rubber Based on Virtual Manufacturing Technology

Development of a Predictive Model for the Thermal Expansion Coefficient of Elastic Disordered... As a novel lightweight metallic material with excellent heat and corrosion resistance, elastic disordered microporous metal rubber (EDMMR) functions as an effective damping and support element in high-temperature environments where traditional polymer rubber fails. In this paper, a multi-scale finite element model for EDMMR is constructed using virtual manufacturing technology ( VMT ). Thermo-mechanical coupling analysis reveals a distinct inward expan- sion and dissipation phenomenon in EDMMR under high-temperature conditions, distinguishing it from porous materials. This phenomenon has the potential to impact the overall dimensions of EDMMR through transmission and accumulation processes. The experimental results demonstrate a random distribution of internal micro springs in EDMMR, considering the contact composition of spring microelements and the pore structure. By incorporating material elasticity, a predictive method for the thermal expansion coefficient of EDMMR based on the Schapery model is proposed. Additionally, standardized processes are employed to manufacture multiple sets of cylindrical EDMMR samples with similar dimensions but varying porosities. Thermal expansion tests are conducted on these samples, and the accuracy of the predicted thermal expansion coefficient is quantitatively validated through residual analysis. This research indicates that EDMMR maintains good structural stability in high-temperature environments. The ther- mal expansion rate of the material exhibits an opposite trend to the variation of elastic modulus with temperature, as the porosity rate changes. Keywords Elastic disordered microporous metal rubber (EDMMR), Virtual manufacturing technology ( VMT ), Elevated temperature, Coefficient of thermal expansion (CTE) 1 Introduction Elastic disordered microporous metal rubber (EDMMR) is a lightweight, eco-friendly material with combined elasticity and damping properties [1–4]. Owing to its unique shape, it is commonly referred to as wire mesh [5] Zhiying Ren and Qinwei Wang contributed equally to this work. or entangled porous wire material [6]. EDMMR is com- posed of various metal wires, including stainless steel [7], *Correspondence: Xianjie Shi copper alloy [8], titanium alloy [9], and others. The man - [email protected] ufacturing process of EDMMR is complex and involves School of Mechanical Engineering and Automation, Institute of Metal several specialized steps such as blank preparation, cold Rubber and Vibration Noise, Fuzhou University, Fuzhou 350116, China Institute of Systems Engineering, China Academy of Engineering stamping forming, winding, stretching, and laying [10]. Physics, Mianyang 621900, Sichuan, China This process results in the formation of metal rubber © The Author(s) 2025. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Ren et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 2 of 14 products with a stable porous prestressed structure and characteristics through experimental methods. Roy et al. exceptional environmental adaptability. These adaptabil - [21] investigated the thermal expansion behavior of inter- ity features include high temperature resistance, radia- penetrating metal/ceramic composites, employing the tion resistance, and oil corrosion resistance, as illustrated Turner model and the Schapery model to determine the in Figure 1. EDMMR finds extensive applications in fields most accurate approximations of the thermal expansion such as vibration and noise reduction, filtration, sealing, coefficients in the longitudinal and transverse directions and other high-end equipment operating under high- at various temperatures. Moreover, EDMMR exhibits temperature environments [11–14]. However, despite its distinct advantages in high-temperature thermal expan- widespread usage, there is a scarcity of research on the sion compared to conventional porous composite materi- thermophysical properties of EDMMR due to its intri- als, thanks to its void accommodation mechanism. cate internal structure. This lack of research leads to On the other hand, temperature variations can alter inadequate prediction accuracy, which hinders its prac- the stress and strain magnitude and distribution, conse- tical implementation. The temperature-induced expan - quently impacting the prestressed state and mechanical sion and contraction of EDMMR not only directly impact properties of EDMMR. Li et  al. [22] developed a con- the macroscopic dimensions of the components but also stitutive model for braided-grooved EDMMR using the influence the local contact state between the wires. Sherwood-Frost equation and considering the tempera- In order to improve this phenomenon, researchers ture effect. They conducted simulations and experiments have conducted investigations into the thermophysical to evaluate the material’s performance under different properties of EDMMR. Lai et al. [15] conducted a series temperature environments. Huang et  al. [23] found that of creep tests to investigate the creep characteristics of temperature increases the friction coefficient of 316L cylindrical EDMMR specimens under high-temperature stainless-steel wire, with higher temperatures soften- and static compression conditions. The findings dem - ing the material and altering the mechanical properties onstrated that the creep resistance of samples at high of EDMMR by promoting oxidation and affecting sur - temperatures can be enhanced through suitable heat face stress. While these studies explore the changes in treatment. Cao et  al. [16] performed high-temperature the macrostructure and material properties of EDMMR tempering on 0Cr18Ni9Ti-based EDMMR, observing within a specific temperature range, further investiga - significant changes in the macroscopic size and material tion is necessary to understand its expansion and defor- properties compared to room temperature. Ma et al. [17] mation mechanisms across a wide temperature range. proposed a method for calculating the thermal expan- Furthermore, numerous scholars from various countries sion coefficient of EDMMR based on the Schapery model have explored the mechanical properties of porous metal for composite materials. Additionally, thermal expansion materials under different temperature conditions, pro - properties of porous materials like foam metal [18, 19] viding valuable insights for EDMMR materials. Povolny have been studied, offering specific reference values for et  al. [24] utilized the material point method to predict high-temperature research on EDMMR. Takezawa et  al. the linear elastic properties of complex microstructures [20] prepared a porous composite material with aniso- in ultra-high-temperature porous ceramics at elevated tropic thermal expansion properties using metal addi- temperatures. Their approach allowed for the analysis tive manufacturing and examined its thermal expansion of the microstructure, large deformation, and internal Figure 1 Preparation process flow and structural characteristics of EDMMR R en et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 3 of 14 contact after elastic behavior in porous ceramics under tangent direction of the baseline, with uniform circum- high temperatures. Additionally, porosity plays a sig- ferential motion. nificant role in the thermophysical properties of porous Next, the finite element model of the EDMMR blank materials [25], which holds great importance for investi- and stamping die is constructed using finite element gating the thermal expansion of EDMMR. analysis software. The chosen material is 304 austenitic In this research, a numerical modeling method known stainless steel, a common material used in EDMMR. The as virtual manufacturing technology (VMT) was uti- stamping forming or unloading process of EDMMR is lized to construct an accurate finite element model that simulated by defining sectional plastic properties, tan - captures the complex internal topological structure of gential modulus, failure strain, and strain rate param- EDMMR. The thermal expansion deformation mecha - eters. Table  1 presents the key parameters for the blank nism of EDMMR across a wide temperature range was stamping process. To optimize computational efficiency, revealed through finite element analysis using a physics- the mesh is divided into dynamic display units, and the based 3D model. Additionally, the influence of tempera - model’s elastic-plastic deformation is calculated using a ture on the relationship between pore structure, thermal classical friction theory and penalty function to generate expansion coefficient, and elastic modulus of EDMMR the final interpenetrating mesh structure. Considering was investigated. The porosity and pore size distribution the symmetrical arrangement of EDMMR, a simplifica - of EDMMR were quantified using the mercury intrusion tion is made by analyzing 1/6 of the circumference of the method and validated using the plane random segmen- EDMMR cylinder sample [26]. tation theory. Finally, experimental investigations on the thermal expansion performance of EDMMR were con- 2.2 Material Parameters and Boundary Conditions ducted using a thermal dilatometer (DIL). Prior to commencing the finite element analysis, it is essential to input the thermophysical parameters of the material. The fundamental thermophysical parameters for the 304 austenitic stainless steel wire are presented in 2 Finite Element Simulation and Analysis Table 2 [27]. For the purpose of the finite element analy - 2.1 VMT of EDMMR sis, the impact of material inhomogeneity on thermo- Reflecting the thermal expansion and deformation of physical parameters is disregarded. the mesoscopic metal wire inside EDMMR accurately To investigate the expansion mechanism of EDMMR and intuitively is challenging using traditional meth- and observe the corresponding outcomes, specific ods, which poses difficulties for research in this area. To boundary conditions are established, as depicted in Fig- overcome this, efficient synchronous VMT is utilized to ure  3. The upper and lower planes serve as the clamp - construct a finite element model that closely resembles ing surface and support surface, respectively, providing the real EDMMR, as illustrated in Figure  2. The process physical constraints in the forming direction. The metal begins by parameterizing the space coordinate curve of wire is subjected to a temperature load ranging from the spiral wire coil, based on the actual EDMMR blank 27.5 °C to 500 °C, with the temperature held constant for winding method. A winding baseline l is established in 5 min/100 °C increment. the global coordinate system, and a moving point o trav- erses along the serpentine baseline. Simultaneously, the 2.3 Finite Element Thermal‑Solid Coupling Analysis coordinates of the spiral coil are determined by moving Based on transient thermal analysis, the temperature field point P in a local coordinate plane perpendicular to the results are utilized as input loads and imported into the static structure module to calculate the thermal defor- mation and thermal stress of the model. As the expan- sion and deformation of EDMMR under a single thermal load result in fretting displacement, the heat generated Table 1 Parameter Settings for VMT of EDMMR Spiral Pitch Mandrel Winding Number Punching diameter p (mm) diameter angle of force D (mm) D (mm) θ (°) winding F (kN) layers Figure 2 Synchronous virtual preparation of the EDMMR finite element model 1.5 1.5 0.5 45 15 5−20 Ren et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 4 of 14 Table 2 Thermophysical parameters of 304 austenitic stainless steel wire Temperature (℃) Density (kg/m ) Thermal conductivity Specific heat (J/kg/K) Coefficient of linear −6 (W/m/K)expansion (10 /K) 25 8020 14.986 495 – 100 7982 16.046 506 16.54 200 7932 17.459 522 17.25 300 7882 18.872 537 17.61 400 7832 20.285 553 17.99 500 7782 21.698 569 18.34 60, 90, and 120 min). With increasing temperature, the deformation of each region of EDMMR gradually inten- sifies. Moreover, the specific pore structure of EDMMR leads to pore inclusion in EDMMR products under high- temperature thermal loads. As the inward deformation expands further, the number of contact pairs between the spiral elements increases, filling the gaps between previ - ously uncontacted pairs. This alteration in contact state expands the volume fraction of the contact pairs. By measuring the deformation of the upper plane Figure 3 Boundary conditions for thermal-solid coupling analysis along the axis and dividing it by the original height of the model, the overall strain rate of the EDMMR model in the forming direction can be obtained, as depicted in Fig- ure 4(b). It can be observed that the size of the EDMMR by friction is nearly negligible compared to the impact model in the molding direction exhibits a consistent out- of ambient temperature. Consequently, the simulation ward extension trend as the temperature increases. This is conducted using sequential coupling, disregarding the trend is beneficial for estimating the expansion amount influence of structural deformation on the temperature and designing the gap size. Additionally, during the tem- field. perature stabilization process, the size of the EDMMR As the ambient temperature rises, the wire material model may undergo slight fluctuations due to material undergoes subtle geometric and volumetric changes. Fig- shrinkage. Moreover, random points near the pore space ure  4(a) illustrates the deformation cloud diagram of the on the EDMMR profile can be selected to analyze the macro model of EDMMR at different time intervals (30, thermal expansion deformation, as shown in Figure  4(c). Figure 4 Outward expansion and inward deformation of EDMMR: (a) Expansion deformation contour maps at different times, (b) Temperature loading mode and overall strain rate of EDMMR, (c) Nodal displacements of porous nodes at different times R en et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 5 of 14 The joint deformation exhibits distinct sectional char - affecting the contact state. The thermal expansion and acteristics. Initially, as the temperature rises (illustrated contact between the wires are interdependent and form in Figure  4(c) stage a  and  b), the displacement of the an interactive dynamic process. To gain a deeper under- joint increases linearly with time. However, as time fur- standing of this phenomenon, a post-processing func- ther elapses (illustrated in Figure  4(c) stage b  and  c), tion based on finite element contact search is utilized to the deformation is accommodated by the pore and thus extract the contact evolution process of the model during suppressed. the heating process, as illustrated in Figure 6. When utilizing the material in a high-temperature The contact density within the model varies due to environment, it is crucial to consider the deformation of differences in relative density. However, the contact the structure size and the resulting thermal stress, tak- behavior gradually strengthens, transitioning from slip ing into account the impact of stress generated by ther- to stick and from distant to near contacts. To quantita- mal expansion on the properties of EDMMR material. tively analyze the dynamic evolution of internal contact To ensure the stability of the EDMMR component dur- in EDMMR, the number of four contact pairs under a ing its usage, it is typically necessary to apply a certain single thermal load was determined using the small ball level of pre-compression. However, due to external con- algorithm. The results are presented in Figure  7. Overall, straints and the influence of its structure, the local area of the number of contact pairs (including slip and adhesive the component cannot fully expand freely when there are contacts) increases by approximately 50% with increas- changes in ambient temperature, resulting in the genera- ing temperature. In contrast, the number of non-contact tion of corresponding stress. Based on the temperature pairs (including close and distant contacts) decreases field distribution results of the EDMMR model combined accordingly. Furthermore, adhesive contact exhibits slow with the model’s displacement, the distribution of equiv- growth, while slip contact dominates and exhibits a sig- alent stress can be calculated, as depicted in Figure 5. nificant increasing trend. Upon further temperature increase, different regions of the model exhibit varying levels of stress distribution, 3 Schapery Model Derivation for EDMMR Thermal with stress levels becoming more pronounced. Previ- Expansion ous studies have indicated that the elastic deformation 3.1 Thermal Dilation of Helical Elements and Their capacity of EDMMR decreases with increasing prestress- Combinations ing [28]. Consequently, elevated temperatures enhance The thermal expansion behavior of EDMMR in high-tem - the stiffness of EDMMR, thereby strengthening its rigid perature environments is influenced by various factors, bearing performance. including material properties and pore microstructure. Therefore, accurately predicting its thermal expan - 2.4 C ontact Form Analysis and Statistical Evaluation sion coefficient is relatively challenging. The simplest The spatial position of the metal wire inside EDMMR approach is to construct a linear model based on the mix- undergoes changes when subjected to loading, thereby ing law, where the bulk thermal expansion coefficient is Figure 5 Stress Contour maps of EDMMR under different temperature environments Ren et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 6 of 14 Figure 6 Contact form change and small ball algorithm linearly related to the thermal expansion coefficients and volume ratios of its components. However, this model neglects the elastic effect of the material during the sim - plification process. While it may yield close predictions to experimental results under specific conditions, it intro - duces significant errors in most cases. In this section, we further investigate the principles of thermal expansion deformation based on helical elements. The structural discretization reveals 211 metal wires in the model, and at the extreme coordinate point of the Y-axis, 625 helical elements can be intercepted, as depicted in Figure 8. The spiral unit undergoes expansion and deforma - tion in all directions upon heating, leading to axial Figure 7 Statistics of metal wire contact quantity Figure 8 Meso-expansion principle of EDMMR model and their equivalent models R en et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 7 of 14 displacement and radial displacement. Based on material corresponding equivalent expansion model, the influence thermodynamics and spring theory, the helical element of the combination form on the thermal expansion can experiences thermal stress due to external constraints, be analyzed, as shown in Figure 8(c). which can be categorized into axial and radial compo- For the uncontacted helical unit group, the pores act as nents [29]: an accommodating mechanism, allowing for the offset of the gap between the units. As a result, only the expansion F = F sin β, of the upper-end portion is transmitted in the molding (1) F = F cos β, direction. In this case, the equivalent thermal expan- sion coefficient α along the forming direction can be where F is the axial component of the helical element described as follows: whose axial angle can be any value, F represents the ther- mal expansion force, F is the radial component, and β is αl sin θ cos β �T the axial angle. α = (l + l sin θ cos β + l)�T 0 1 Considering that the helical element does not experi- (5) l sin θ cos β ence twisting or bending during the expansion process, = α ≤ α, 2(l + l sin θ cos β + l) 0 1 the effects of torque and bending moment can be dis - regarded. By applying Castigliano’s theorem [30], the where α represents the thermal expansion coefficient of deformation of the helical element can be determined as the stainless steel wire, l denotes the original length of the follows: spiral unit, and ∆T represents the temperature change. l Meanwhile, the lower end of the uncontacted helical  ∂F F 4FD sin θ t1 t1 L �Z = · ds = ,  unit group remains unaffected by forces during expan - ∂F EA Ed cos θ sion, resulting in an equivalent stiffness given by: � (2) ∂F F 2FD cos θ cos β  t2 t2 L  �R = · ds = , 2 k = k(β ). 1 1 (6) ∂F EA Ed For the helical element group in sliding contact, where ∆Z represents the expansion deformation of the the equivalent thermal expansion coefficient can be helical unit under axial load, ∆R represents the expan - expressed as follows: sion deformation of the helical unit under radial load, D is the diameter of the helical unit, θ is the helix angle αl sin α cos β �T + αl sin α cos β �T 1 2 α = = α. of the helical unit, β is the axial angle of the helical unit, 2 (l sin α cos β +l sin α cos β )�T 1 2 F = F sin θ , F = F sin ϕ cos θ = F sin ϕ cos β cos θ , t1 t2 r (7) E is the elastic modulus of the material, and d rep- At this stage, the helical element generates both a sup- resents the wire diameter of the coil, A = πd /4 , porting force and a reaction force. The equivalent stiff - l 2π ds = D/2 cos αdθ. 0 0 ness of the helical element can be expressed as [31]: The comprehensive deformation of the helical unit along the forming direction can be obtained as follows: k(β )k(β ) 1 2 k = . sin (|β − β | ) 1 2 1 + tan (|β − β | ) k(β ) + k(β ) 1 2 1 2 �n=�Zcosβ + �R sin β. cos (|β − β | ) (3) 1 2 (8) Then, the stiffness of the helical element along the Similarly, the equivalent thermal expansion coefficient forming direction can be calculated for any axis angle: of the helical element group in the extrusion contact rela- tionship can be determined as follows: F Ed cos θ k(β) = = , �n 4D cos β(ω (θ) + D cos β sin β(ω (θ)) 1 2 αl sin α cos β �T + αl sin α cos β �T 1 2 (4) α = = α. (l sin α cos β +l sin α cos β )�T 1 2 where ω and ω represent the structural deformation 1 2 (9) parameters of the thermal expansion of the helical ele- At this stage, the combined form of the helical unit 2 2 ment, ω = sin θ, ω = cos θ. 1 2 group resembles a parallel structure, resulting in the fol- In addition to the structural parameters of the spiral lowing equivalent stiffness: element itself, the combination form between the ele- ments also influences the thermal expansion of the overall k = k(β ) + k(β ). 3 1 2 (10) model. Therefore, representative combination forms can uTh s, the elastic modulus of the structure can be be summarized based on the different relative positions expressed by its stiffness: and contact states of the helical units. By establishing the Ren et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 8 of 14 The proportion of uncontacted pairs is the largest in E = k A, i = 1, 2, 3, (11) i i EDMMR materials at low temperatures due to the pres- where E (i = 1, 2, 3) represents the elastic moduli of the ence of pores. Consequently, the uncontacted pairs have three fundamental combinations of helical unit groups. the most significant influence on the thermal expansion It can be observed that the thermal expansion of the coefficient of EDMMR. As the temperature increases, uncontacted helical unit is constrained by the presence of the thermal expansion deformation gradually intensi- gaps, limiting its transfer from forming to thermal expan- fies. The thermal expansion deformation, resulting from sion. On the other hand, the contacting helical unit directly external constraints and material expansion, affects the transfers the deformation from forming to thermal expan- gaps within EDMMR, leading to changes in its internal sion through contact, unaffected by other factors. contact state. Additionally, Figure 9 presents the thermal expansion of EDMMR samples at different temperatures 3.2 P rediction of Thermal Expansion Coefficient based on the Schapery model of finite elements and the Through the derivation of the above formula, it can be thermal expansion behavior of EDMMR. observed that the equivalent expansion coefficient and Based on the analysis of the thermal expansion perfor- elastic modulus of any helical unit group are closely mance of EDMMR, it is observed that the thermal expan- related to its axis angles β and β . In this regard, the axis sion coefficient of each component is influenced by the 1 2 angle distribution of the spiral units inside the model is volume fraction and elastic modulus of the combined calculated based on structural discrete statistics. On this form. The equivalent thermal expansion coefficient of the basis, a random distribution combination of metal wires uncontacted helical unit group is correlated with the rela- at any angle is established to capture the complex and tive density of EDMMR. Similarly, for the spiral element disordered discontinuous structure of metal wires within group in sliding and adhesive contact, the thermal expan- the EDMMR. sion coefficient is solely influenced by the thermophysi - The Schapery model [32], commonly used in woven cal properties of the EDMMR material. The macroscopic composite materials, provides a description of the influ - thermal expansion performance of EDMMR is primar- ence of each component’s elastic modulus on the overall ily influenced by its structure. Furthermore, the volume thermal expansion coefficient based on the energy prin - fraction of each component in EDMMR is directly related ciple. In the case of EDMMR, the various contact forms to the relative density of the material. exhibit different effects on the thermal expansion coef - ficient, similar to materials with different components. 4 Experimental Verification Additionally, the thermal expansion performance, vol- 4.1 Experimental Instrumentation and Principles ume fraction, and elastic modulus of EDMMR vary under EDMMR is composed of metal wires surrounded by different relative densities, resulting in different con - pores filled with a gaseous medium. The pore structure tributions to the macro thermal expansion of EDMMR. has varying degrees of impact on the material’s macro- Considering the impact of different contact forms on the scopic properties, including thermal expansion proper- thermal expansion behavior of EDMMR, it can be con- ties. The pore structure of EDMMR is characterized by cluded that: parameters such as porosity, shape, and pore size distri- bution. Typically, the internal pores of EDMMR exhibit N N α = α ϕ E ϕ E , e i i i i i (12) 1 1 where α represents the equivalent thermal expansion coefficient of the entire structure, and α , E , and ϕ denote i i i the thermal expansion coefficient, elastic modulus, and volume fraction of the respective components. By substituting the derived volume fraction, equiva- lent thermal expansion coefficient, and equivalent elastic modulus of the different combinations of helical element groups into the following formula, the Schapery model for EDMMR can be established, enabling the calculation of its equivalent thermal expansion coefficient: α ϕ E + α ϕ E + α ϕ E 1 1 1 2 2 2 3 3 3 α = . MR (13) ϕ E + ϕ E + ϕ E 1 1 2 2 3 3 Figure 9 Prediction of thermal expansion result of EDMMR R en et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 9 of 14 random distribution and irregular shapes, making it chal- Table 3 Parameters of different sample groups lenging to characterize the structure using conventional Samples serial Average Average Average methods. However, an equivalent diameter can be calcu- number height (mm) diameter (mm) relative density (%) lated using a simplified model. To investigate the pore structure of EDMMR, elec- R–1 8.00 8.00 – tron microscopy was used to observe the internal pores. M–4 8.02 8.01 29.8 Porosity and pore size distribution of the EDMMR sam- M–5 8.03 8.02 40.3 ples were measured using the automatic mercury injec- M–6 8.05 8.01 50.4 tion instrument MAC AutoPore IV 9500, as shown in Figure 10(a). The measurement is based on the Washburn equation [33], which relates the pore size to the mercury parameters presented in Table  3. The experimental pro - contact angle with the material and the applied mercury cedure involved placing different sample groups onto the pressure. sample holder and enclosing them within a heating fur- The thermal expansion experiment of EDMMR uti - nace. The ambient temperature was gradually raised to lized the DIL 402C thermal dilatometer manufactured by the desired temperature using a thermocouple as the heat NETZSCH, Germany. The experimental setup includes source, with a heating rate of 5 ℃/min [34]. Due to the a temperature-controlled furnace, a test tube sample material’s efficient heat transfer capability and the small holder, a displacement sensor, and other components, size of the samples, which ensured prolonged insulation as shown in Figure  10(b). The thermal dilatometer offers time, the temperature difference between the samples excellent linearity and resolution (0.125 μm, 0.1 K) within and the environment could be disregarded. Therefore, the a wide measurement range (−5000 μm to +5000 μm) and temperature inside the furnace was considered equiva- temperature control range (−180 ℃ to 2000 ℃). lent to the sample temperature. Additionally, high-purity During the experiment, the sample is subjected to ther- argon gas was introduced at a flow rate of 50 mL/min as mal expansion by the heating unit, causing the pusher to a protective gas to maintain uniform sample temperature move along the linear guide of the sample container. The distribution, prevent oxidation, and ensure a constant displacement sensor, connected to an optical encoder, pressure. Finally, the expansion data of the samples were directly measures the corresponding length change periodically measured and recorded. on the appropriate scale. For clarity, the sample, heat- ing unit, pusher, linear guide, and displacement sensor 4.2 Experimental Results and Discussion are color-coded in Figure  10(b) for easier visualization: The density of the wire and the pore size of the EDMMR purple, red, green, blue, and yellow, respectively. Impor- sample are directly affected by different porosity. Scan - tantly, the binding force between the push rod and the ning electron microscope (SEM) was used to observe the sample is negligible during this process, allowing for EDMMR samples with varying porosity, as shown in Fig- thermal expansion in a free state. ure 11. It can be observed that the pores surrounding the To investigate the structural stability of thermal metal wire exhibit an overall irregular geometric shape. expansion in EDMMR, EDMMR samples of the same With an increase in porosity, the arrangement of wires size were included as a control group, with sample inside the sample becomes more sparse. Figure 10 Pore structure and thermal expansion experiments of EDMMR: (a) Mercury intrusion measuring instrument, (b) Thermal dilatometer and experimental schematic Ren et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 10 of 14 Figure 11 SEM images of the EDMMR samples with different porosity levels Figure 13 Variation curve of thermal expansion of EDMMR with Temperature the pores can accommodate the inward expansion of the metal wire, thereby enhancing the structural sta- bility of the EDMMR component in high-temperature environments. The thermal expansion test was conducted on the Figure 12 Mercury intrusion experiment curves samples within a temperature range of 25 ℃ to 700 ℃. The measurement focused on the thermal expansion deformation of EDMMR during the heating process. The variation of the linear expansion coefficient in the Table 4 Experimental pore size distribution results forming direction with temperature was calculated and Sample porosity Aperture concentration Immersion recorded. The results are presented in Figure  13. Com- (%) distribution value (mm) amount −1 paring the change curve of the rubber material (used as (g·mL ) a control) with that of EDMMR, it can be observed that 30 0.142 0.380 the EDMMR exhibits better structural stability. Initially, 40 0.247 0.631 the curve shows nearly linear growth as the temperature 50 0.288 1.014 increases. However, at specific temperatures (287.5  ℃, 514.0 ℃, 615.1 ℃, and 635.7 ℃), the sample encounters phase transformation softening points. This leads to var - The equivalent pore size inside the samples with ying degrees of shrinkage due to the relaxation of internal different porosity was measured using the mercury strain, which differs from the behavior observed in other intrusion method. Figure  12 and Table  4 present the materials. In the case of entropy elastic material such as mercury intrusion curves and pore size distribution rubber, the softening shrinkage can be thermodynami- results of the EDMMR samples. The mercury intru - cally explained as a shrinkage occurring under constant sion curves exhibit symmetry around their mean val- tension as the temperature increases. The rate of thermal ues and approximate a normal distribution, which has shrinkage can be expressed as: been verified by previous studies using the plane ran - ∂L L 1 dom segmentation theory [35]. Moreover, as the poros- =−f · · , (14) ity increases across different samples, the average pore ∂T T E size also increases. By comparing the minimum aper- where f is the tension per unit area, T is the absolute ture concentration distribution value (0.142  mm) with temperature during the thermal expansion test, L is the the maximum deformation result (0.096 mm) obtained length of the sample in the load direction, L is the initial from the finite element simulation, it is evident that length of the sample in the load direction, and E is the the latter is smaller than the former. This indicates that elastic modulus. R en et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 11 of 14 The elastic component of EDMMR primarily exhibits show good agreement, indicating that these coefficients energy elasticity, which results in its temperature tol- are inherent properties of the material. Furthermore, erance and wider range of temperature stability. It can the physical expansion coefficient remains relatively maintain a stable bearing capacity, characterized by stiff - constant until a specific softening point, demonstrating ness and hardness, even in high-temperature environ- the expansion stability of the EDMMR sample at high ments. Only when the temperature exceeds its limit, its temperatures. performance will be weakened, although the reversible deformation value remains small. Furthermore, the slope 4.3 Residual Analysis of the linear phase in the EDMMR curve is influenced by Residual analysis was conducted by comparing the pre- the porosity of the sample and increases as the porosity dicted thermal expansion with the experimental results, decreases. as presented in Figure 15. The corresponding data can be Furthermore, the sample’s physical expansion coeffi - found in Table 5. cient and engineering expansion coefficient can be deter - Based on the preceding analysis, the residual points mined as follows: in the horizontal band area are evenly distributed, dem- onstrating the effectiveness of the constructed predic - 1 dL tion model in capturing the expansion characteristics of α(T ) = · , (15) L dT EDMMR. To further assess the model’s validity, the cor- relation index R can be calculated using the residual sum of squares (RSS) and the total sum of squares (TSS). �L �L 1 α = − · , (T −T ) 1 2 L L T − T 1 1 2 1 (T ) (T ) 2 1 RSS = (y − yˆ ) , i i (17) (16) i=1 where dL/dT represents the partial derivative of length with respect to temperature, ∆L is the sample elongation, ∆T is the temperature change, L is the original length of 2 TSS = (y − y) , i (18) the sample, T is the fixed reference temperature (22  ℃), i=1 and T is the maximum heating temperature. The instantaneous change rate of the linear size of the RSS EDMMR sample in the forming direction is represented R = 1 − , (19) TSS by the former, while the latter represents the average change rate of the linear length of the sample within a where, n is the number of sample observations, y is specific temperature range (T , T ). The change curves the sample observation value, y ˆ is the model predic- 1 2 of these results with temperature are shown in Fig- tion value, y represents the average value of sample ure 14(a) and (b), respectively. The physical and engineer - observations. ing expansion coefficient curves of the EDMMR samples Figure 14 Graphs depict the curves of different expansion coefficients for EDMMR Ren et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 12 of 14 Figure 15 Residual analysis of the studied EDMMR samples Table 5 Experimental results and predicted values of thermal expansion of samples at different temperatures (the predicted values in parentheses) −3 −3 −3 Temperature (℃)M–4 (10 )M–5 (10 )M–6 (10 ) 100 2.53 (2.31) 2.20 (2.00) 0.82 (1.30) 200 6.60 (6.93) 5.80 (6.10) 3.90 (4.30) 300 10.6 (10.4) 9.50 (9.10) 7.30 (6.80) 400 14.5 (14.4) 13.8 (13.3) 10.9 (10.3) 500 18.2 (18.0) 17.8 (17.2) 12.5 (11.9) (1) The internal topological structure of EDMMR was Table 6 Accuracy analysis of the model effectively reconstructed using VDT, followed by Sample serial RSS TSS R finite element thermal-solid coupling analysis. The number results showed that with the increasing tempera- −7 −4 M–4 2.66×10 2.47×10 0.9892 ture, the stress in the material became concentrated −7 −4 M–5 9.00×10 2.34×10 0.9616 at both ends and gradually increased, leading to a −6 −4 M–6 1.36×10 1.35×10 0.8994 more significant rigid support performance. Fur - thermore, EDMMR exhibited enhanced struc- tural stability compared to other porous materials, thanks to its unique pore accommodation mecha- The results are presented in Table  6, indicating the cor- nism. relation indices R for the three samples. A higher value (2) The thermal expansion behavior of EDMMR is of R indicates greater prediction accuracy and a stronger mainly governed by the interactions among the linear correlation between the observed and predicted contacting helical coils. Furthermore, the thermal variables. It can be observed that R values for all three expansion properties, volume fractions, and elas- samples are close to 1. However, as the porosity increases, tic moduli of the individual components within the irregularity of the samples also increases, resulting in EDMMR exhibit variations, leading to different higher errors. contributions to the overall macroscopic thermal expansion. The transition between gaps among 5 Conclusions uncontacted microelements and different contact This study conducted finite element simulation, theo - forms plays a crucial role in determining the ther- retical analysis, and experimental verification to investi - mal expansion coefficient of EDMMR under vary - gate the thermal expansion deformation mechanism of ing relative densities. EDMMR at high temperatures. The research methodol - (3) A modified prediction method was developed for ogy is described as follows: the thermal expansion coefficient of EDMMR, incorporating structural discretization and random distribution of contacts. This method considers the R en et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 13 of 14 [9] B Gadot, O R Martinez, S R Du Roscoat, et al. Entangled single-wire NiTi macroscopic parameters of EDMMR and allows for material: A porous metal with tunable superelastic and shape memory accurate prediction of its thermal expansion perfor- properties. Acta Materialia, 2015: 311-323 mance under different temperatures, considering [10] Y Yang, Z Y Ren, S Y Zhao, et al. One-step fabrication of thermal resistant, corrosion resistant metal rubber for oil/water separation. Colloids and various preparation and material parameters. The Surfaces A: Physicochemical and Engineering Aspects, 2019: 157-164 results demonstrate a positive correlation between [11] T Li, H B Bai, X Xue, et al. Fatigue properties of knitted-dapped metal rub- relative density and temperature in relation to the bers under high temperature environment. China Mechanical Engineering, 2019, 30(9): 9. (in Chinese) thermal expansion behavior of EDMMR. [12] Z Y Ren, W Gang, Z G Guo. Biomimetic high-intensity superhydrophobic metal rubber with anti-corrosion property for industrial oil-water separa- tion. New Journal of Chemistry, 2018, 43(4). Acknowledgements [13] Z Y Ren, D D Wu, L W Shi, et al. Research on the structural performance of Not applicable. large aspect ratio O-ring metal rubber seals based on virtual preparation. Lubrication Engineering, 2023, 48: 40-47. (in Chinese) Authors’ Contributions [14] Z N Zhang, N Yin, S Chen, et al. Tribo-informatics: Concept, architecture, ZR and XS were responsible for formal analysis, visualization, and editing; and case study. Friction, 2021, 9(3): 14. QW contributed to the investigation, conceptualization, and conducting the [15] F Q Lai, X F Hao, N N Liu, Y, et al. Creep properties of cylinder metal rubber experiments; RF and ZH provided assistance with methodology and resources. under static compression at elevated temperatures. Symmetry, 2023, All authors read and approved the final manuscript. 15(2). [16] F L Cao, H B Bai, G Q Ren, et al. Constitutive model of metal rubber Funding material based on curved cantilever beam of variable length. Journal of Supported by National Natural Science Foundation of China (Grant Nos. Mechanical Engineering, 2012, 48(24): 61. U2330202, 52175162, 51805086, 51975123), Fujian Provincial Technological [17] Y H Ma, X L Tong, B Zhu, et al. Theoretical and experimental study on Innovation Key Research and Industrialization Projects (Grant Nos. 2023XQ005, thermophysical properties of metal rubber. Journal of Physics, 2013, (4): 2024XQ010), Project of Guangdong Provincial Science and Technology Bureau 10. (in Chinese) of Jiangmen City (Grant No. 2023780200030009506). [18] H Ye, M Y Ma, J L Yu. Anomalies in mid-high-temperature linear thermal expansion coefficient of the closed-cell aluminum foam. Science Bulletin: Data availability English version, 2014: 3669–3675. The data that support the findings of this study are available on request from [19] S Chen, J Marx, A Rabiei. Experimental and computational studies on the the corresponding author, [Shi], upon reasonable request. thermal behavior and fire retardant properties of composite metal foams. International Journal of Thermal Sciences, 2016: 106–170. [20] A Takezawa, M Kobashi. Design methodology for porous composites Declarations with tunable thermal expansion produced by multi-material topology optimization and additive manufacturing. Composites Part B: Engineering, Competing Interests 2017, 131: 21-29. The authors declare no competing financial interests. [21] S Roy, A Nagel, K A Weidenmann. Anisotropic thermal expansion behav- ior of an interpenetrating metal/ceramic composite. Thermochimica Acta, 2020, 684. Received: 27 February 2023 Revised: 19 December 2024 Accepted: 23 [22] T Li, H B Bai, F L Cao. A quasi-static compression constitutive model December 2024 for knitted-dapped metal rubber considering temperature effect. Acta Aeronautica et Astronautica Sinica, 2018, 39(10). (in Chinese) [23] M J Huang, Y L Fu, X X Qiao, et al. investigation into friction and wear characteristics of 316L stainless-steel wire at high temperature. Materials, 2023, 16(1). References [24] S J Povolny, G D Seidel, C Tallon. Investigating the mechanical behavior [1] Z Y Ren, R Z Fang, X C Chen, et al. Study on anisotropic constitutive prop- of multiscale porous ultra-high temperature ceramics using a quasi-static erties of metal rubber based on virtual fabrication technology. Journal of material point method. Mechanics of Materials, 2021, 160(3): 103976. Mechanical Engineering, 2021, 57(24): 211-222. [25] J Y Kong, H Y Zou, K Zeng, et al. Investigation of optimization methods [2] L W Shi, Z Y Ren, Z H Huang, et al. Research on trajectory optimization of for metal foam with two-dimensional porosity gradient in shell-and-tube multilateral thin metal rubber automatic laying based on virtual fabrica- latent heat storage. Journal of Energy Storage, 2023, 63: 107004. tion technology. Advanced Engineering Materials, 2021, 24(2). [26] Ren Z, Shen L, Huang Z, et al. Study on multi-point random contact [3] C H Zhou, Z Y Ren, Y X Lin, et al. Hysteresis dynamic model of metal rub- characteristics of metal rubber spiral mesh structure. IEEE Access, 2019, 7: ber based on higher-order nonlinear friction (HNF). Mechanical Systems 132694-132710. and Signal Processing, 2023: 189. [27] K C Mills. Recommended values of thermophysical properties for selected [4] P Yang, H B Bai, X Xue, et al. Vibration reliability characterization and commercial alloys. Woodhead Publishing, 2002. damping capability of annular periodic metal rubber in the non-molding [28] C Kartik, R Jem, C Elizabeth. Mechanical behaviour of tangled metal wire direction. Mechanical Systems and Signal Processing, 2019, 132: 622-639. devices. Mechanical Systems and Signal Processing, 2019, 118: 13-29. [5] X Xue, Y H Wei, Fang W, et al. Fabrication technology and shear failure [29] Q W Wang, Z Y Ren, L W Shi, et al. Stiffness constitutive characteristics of behaviours of elastic–porous sandwich structure with entangled metallic elastic disordered microporous metal rubber considering temperature wire mesh. Thin-Walled Structures. 2022, 170: 108599. effects. Advanced Engineering Materials, 2023, 25(23). [6] Y Zhu, Y W Wu, H B Bai, et al. Research on vibration reduction design of [30] W P Hu, Q C Meng. Understand the application of Karnaugh’s theorem foundation with entangled metallic wire material under high tempera- from the principle of virtual work. Mechanics in Engineering, 2019, 41(4): 4. ture. Shock and Vibration, 2019: 1-16. [31] B Zhu, Y H Ma, D Y Zhang, et al. A constitutive model of metal rubber [7] H Y Li, Z Y Ren, X L Su, et al. Study on the fretting wear evolution model of based on hysteresis property. Journal of Physics, 2012, 61(7): 8. wires with curvature inside metal rubber. Tribology Letters, 2023, 71(1). [32] R A Schapery. Thermal expansion coefficients of composite materials [8] W Zhang, X Xue, H B Bai. Mechanical and electrical properties of Cu-steel based on energy principles. Journal of Composite Materials, 1968, 2(3): bimetallic porous composite with a double-helix entangled structure. 380-404. Compos. Struct., 2021, 255: 112886 [33] E W Washburn. The dynamics of capillary flow. Physical Review Journals Archive, 1921, 17(3): 273-283. Ren et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 14 of 14 [34] J Jin, W Bao, P Zhang, et al. Creep property of TMCP high-strength steel Q690CFD at elevated temperatures. Journal of Materials in Civil Engineer- ing, 2020, 32(2): 4019361-4019364. [35] Y G Guo, Y H Xia, Z B Chen, et al. Study on pore size distribution charac- teristics of metal rubber filter materials. Journal of Filtration & Separation, 2008. (in Chinese) Zhiying Ren born in 1980, is currently a professor and doctoral supervisor at School of Mechanical Engineering and Automation, Fuzhou University, China. She received her PhD degree from Fuzhou University, China, in 2015. Her research interests include the research of metal rubber materials and equipment vibration and noise reduc- tion technology. Qinwei Wang born in 1999, is currently a master candidate at Institute of Metal Rubber and Vibration Noise, School of Mechanical Engi- neering and Automation, Fuzhou University, China. Rongzheng Fang born in 1997, is currently a master candidate at Institute of Metal Rubber and Vibration Noise, School of Mechanical Engi- neering and Automation, Fuzhou University, China. Zihao Huang born in 1998, is currently a master candidate at Insti- tute of Metal Rubber and Vibration Noise, School of Mechanical Engi- neering and Automation, Fuzhou University, China. Xianjie Shi born in 1985, is currently a senior engineer at Institute of Systems Engineering, China Academy of Engineering Physics, China. He received his PhD degree in mechanical design and theory from Harbin Engineering University, China, in 2014. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Chinese Journal of Mechanical Engineering Springer Journals

Development of a Predictive Model for the Thermal Expansion Coefficient of Elastic Disordered Microporous Metal Rubber Based on Virtual Manufacturing Technology

Download PDF
14 pages

Loading next page...
 
/lp/springer-journals/development-of-a-predictive-model-for-the-thermal-expansion-bM0ox5WfaC

References (32)

Publisher
Springer Journals
Copyright
Copyright © The Author(s) 2025
ISSN
1000-9345
eISSN
2192-8258
DOI
10.1186/s10033-024-01165-8
Publisher site
See Article on Publisher Site

Abstract

As a novel lightweight metallic material with excellent heat and corrosion resistance, elastic disordered microporous metal rubber (EDMMR) functions as an effective damping and support element in high-temperature environments where traditional polymer rubber fails. In this paper, a multi-scale finite element model for EDMMR is constructed using virtual manufacturing technology ( VMT ). Thermo-mechanical coupling analysis reveals a distinct inward expan- sion and dissipation phenomenon in EDMMR under high-temperature conditions, distinguishing it from porous materials. This phenomenon has the potential to impact the overall dimensions of EDMMR through transmission and accumulation processes. The experimental results demonstrate a random distribution of internal micro springs in EDMMR, considering the contact composition of spring microelements and the pore structure. By incorporating material elasticity, a predictive method for the thermal expansion coefficient of EDMMR based on the Schapery model is proposed. Additionally, standardized processes are employed to manufacture multiple sets of cylindrical EDMMR samples with similar dimensions but varying porosities. Thermal expansion tests are conducted on these samples, and the accuracy of the predicted thermal expansion coefficient is quantitatively validated through residual analysis. This research indicates that EDMMR maintains good structural stability in high-temperature environments. The ther- mal expansion rate of the material exhibits an opposite trend to the variation of elastic modulus with temperature, as the porosity rate changes. Keywords Elastic disordered microporous metal rubber (EDMMR), Virtual manufacturing technology ( VMT ), Elevated temperature, Coefficient of thermal expansion (CTE) 1 Introduction Elastic disordered microporous metal rubber (EDMMR) is a lightweight, eco-friendly material with combined elasticity and damping properties [1–4]. Owing to its unique shape, it is commonly referred to as wire mesh [5] Zhiying Ren and Qinwei Wang contributed equally to this work. or entangled porous wire material [6]. EDMMR is com- posed of various metal wires, including stainless steel [7], *Correspondence: Xianjie Shi copper alloy [8], titanium alloy [9], and others. The man - [email protected] ufacturing process of EDMMR is complex and involves School of Mechanical Engineering and Automation, Institute of Metal several specialized steps such as blank preparation, cold Rubber and Vibration Noise, Fuzhou University, Fuzhou 350116, China Institute of Systems Engineering, China Academy of Engineering stamping forming, winding, stretching, and laying [10]. Physics, Mianyang 621900, Sichuan, China This process results in the formation of metal rubber © The Author(s) 2025. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Ren et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 2 of 14 products with a stable porous prestressed structure and characteristics through experimental methods. Roy et al. exceptional environmental adaptability. These adaptabil - [21] investigated the thermal expansion behavior of inter- ity features include high temperature resistance, radia- penetrating metal/ceramic composites, employing the tion resistance, and oil corrosion resistance, as illustrated Turner model and the Schapery model to determine the in Figure 1. EDMMR finds extensive applications in fields most accurate approximations of the thermal expansion such as vibration and noise reduction, filtration, sealing, coefficients in the longitudinal and transverse directions and other high-end equipment operating under high- at various temperatures. Moreover, EDMMR exhibits temperature environments [11–14]. However, despite its distinct advantages in high-temperature thermal expan- widespread usage, there is a scarcity of research on the sion compared to conventional porous composite materi- thermophysical properties of EDMMR due to its intri- als, thanks to its void accommodation mechanism. cate internal structure. This lack of research leads to On the other hand, temperature variations can alter inadequate prediction accuracy, which hinders its prac- the stress and strain magnitude and distribution, conse- tical implementation. The temperature-induced expan - quently impacting the prestressed state and mechanical sion and contraction of EDMMR not only directly impact properties of EDMMR. Li et  al. [22] developed a con- the macroscopic dimensions of the components but also stitutive model for braided-grooved EDMMR using the influence the local contact state between the wires. Sherwood-Frost equation and considering the tempera- In order to improve this phenomenon, researchers ture effect. They conducted simulations and experiments have conducted investigations into the thermophysical to evaluate the material’s performance under different properties of EDMMR. Lai et al. [15] conducted a series temperature environments. Huang et  al. [23] found that of creep tests to investigate the creep characteristics of temperature increases the friction coefficient of 316L cylindrical EDMMR specimens under high-temperature stainless-steel wire, with higher temperatures soften- and static compression conditions. The findings dem - ing the material and altering the mechanical properties onstrated that the creep resistance of samples at high of EDMMR by promoting oxidation and affecting sur - temperatures can be enhanced through suitable heat face stress. While these studies explore the changes in treatment. Cao et  al. [16] performed high-temperature the macrostructure and material properties of EDMMR tempering on 0Cr18Ni9Ti-based EDMMR, observing within a specific temperature range, further investiga - significant changes in the macroscopic size and material tion is necessary to understand its expansion and defor- properties compared to room temperature. Ma et al. [17] mation mechanisms across a wide temperature range. proposed a method for calculating the thermal expan- Furthermore, numerous scholars from various countries sion coefficient of EDMMR based on the Schapery model have explored the mechanical properties of porous metal for composite materials. Additionally, thermal expansion materials under different temperature conditions, pro - properties of porous materials like foam metal [18, 19] viding valuable insights for EDMMR materials. Povolny have been studied, offering specific reference values for et  al. [24] utilized the material point method to predict high-temperature research on EDMMR. Takezawa et  al. the linear elastic properties of complex microstructures [20] prepared a porous composite material with aniso- in ultra-high-temperature porous ceramics at elevated tropic thermal expansion properties using metal addi- temperatures. Their approach allowed for the analysis tive manufacturing and examined its thermal expansion of the microstructure, large deformation, and internal Figure 1 Preparation process flow and structural characteristics of EDMMR R en et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 3 of 14 contact after elastic behavior in porous ceramics under tangent direction of the baseline, with uniform circum- high temperatures. Additionally, porosity plays a sig- ferential motion. nificant role in the thermophysical properties of porous Next, the finite element model of the EDMMR blank materials [25], which holds great importance for investi- and stamping die is constructed using finite element gating the thermal expansion of EDMMR. analysis software. The chosen material is 304 austenitic In this research, a numerical modeling method known stainless steel, a common material used in EDMMR. The as virtual manufacturing technology (VMT) was uti- stamping forming or unloading process of EDMMR is lized to construct an accurate finite element model that simulated by defining sectional plastic properties, tan - captures the complex internal topological structure of gential modulus, failure strain, and strain rate param- EDMMR. The thermal expansion deformation mecha - eters. Table  1 presents the key parameters for the blank nism of EDMMR across a wide temperature range was stamping process. To optimize computational efficiency, revealed through finite element analysis using a physics- the mesh is divided into dynamic display units, and the based 3D model. Additionally, the influence of tempera - model’s elastic-plastic deformation is calculated using a ture on the relationship between pore structure, thermal classical friction theory and penalty function to generate expansion coefficient, and elastic modulus of EDMMR the final interpenetrating mesh structure. Considering was investigated. The porosity and pore size distribution the symmetrical arrangement of EDMMR, a simplifica - of EDMMR were quantified using the mercury intrusion tion is made by analyzing 1/6 of the circumference of the method and validated using the plane random segmen- EDMMR cylinder sample [26]. tation theory. Finally, experimental investigations on the thermal expansion performance of EDMMR were con- 2.2 Material Parameters and Boundary Conditions ducted using a thermal dilatometer (DIL). Prior to commencing the finite element analysis, it is essential to input the thermophysical parameters of the material. The fundamental thermophysical parameters for the 304 austenitic stainless steel wire are presented in 2 Finite Element Simulation and Analysis Table 2 [27]. For the purpose of the finite element analy - 2.1 VMT of EDMMR sis, the impact of material inhomogeneity on thermo- Reflecting the thermal expansion and deformation of physical parameters is disregarded. the mesoscopic metal wire inside EDMMR accurately To investigate the expansion mechanism of EDMMR and intuitively is challenging using traditional meth- and observe the corresponding outcomes, specific ods, which poses difficulties for research in this area. To boundary conditions are established, as depicted in Fig- overcome this, efficient synchronous VMT is utilized to ure  3. The upper and lower planes serve as the clamp - construct a finite element model that closely resembles ing surface and support surface, respectively, providing the real EDMMR, as illustrated in Figure  2. The process physical constraints in the forming direction. The metal begins by parameterizing the space coordinate curve of wire is subjected to a temperature load ranging from the spiral wire coil, based on the actual EDMMR blank 27.5 °C to 500 °C, with the temperature held constant for winding method. A winding baseline l is established in 5 min/100 °C increment. the global coordinate system, and a moving point o trav- erses along the serpentine baseline. Simultaneously, the 2.3 Finite Element Thermal‑Solid Coupling Analysis coordinates of the spiral coil are determined by moving Based on transient thermal analysis, the temperature field point P in a local coordinate plane perpendicular to the results are utilized as input loads and imported into the static structure module to calculate the thermal defor- mation and thermal stress of the model. As the expan- sion and deformation of EDMMR under a single thermal load result in fretting displacement, the heat generated Table 1 Parameter Settings for VMT of EDMMR Spiral Pitch Mandrel Winding Number Punching diameter p (mm) diameter angle of force D (mm) D (mm) θ (°) winding F (kN) layers Figure 2 Synchronous virtual preparation of the EDMMR finite element model 1.5 1.5 0.5 45 15 5−20 Ren et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 4 of 14 Table 2 Thermophysical parameters of 304 austenitic stainless steel wire Temperature (℃) Density (kg/m ) Thermal conductivity Specific heat (J/kg/K) Coefficient of linear −6 (W/m/K)expansion (10 /K) 25 8020 14.986 495 – 100 7982 16.046 506 16.54 200 7932 17.459 522 17.25 300 7882 18.872 537 17.61 400 7832 20.285 553 17.99 500 7782 21.698 569 18.34 60, 90, and 120 min). With increasing temperature, the deformation of each region of EDMMR gradually inten- sifies. Moreover, the specific pore structure of EDMMR leads to pore inclusion in EDMMR products under high- temperature thermal loads. As the inward deformation expands further, the number of contact pairs between the spiral elements increases, filling the gaps between previ - ously uncontacted pairs. This alteration in contact state expands the volume fraction of the contact pairs. By measuring the deformation of the upper plane Figure 3 Boundary conditions for thermal-solid coupling analysis along the axis and dividing it by the original height of the model, the overall strain rate of the EDMMR model in the forming direction can be obtained, as depicted in Fig- ure 4(b). It can be observed that the size of the EDMMR by friction is nearly negligible compared to the impact model in the molding direction exhibits a consistent out- of ambient temperature. Consequently, the simulation ward extension trend as the temperature increases. This is conducted using sequential coupling, disregarding the trend is beneficial for estimating the expansion amount influence of structural deformation on the temperature and designing the gap size. Additionally, during the tem- field. perature stabilization process, the size of the EDMMR As the ambient temperature rises, the wire material model may undergo slight fluctuations due to material undergoes subtle geometric and volumetric changes. Fig- shrinkage. Moreover, random points near the pore space ure  4(a) illustrates the deformation cloud diagram of the on the EDMMR profile can be selected to analyze the macro model of EDMMR at different time intervals (30, thermal expansion deformation, as shown in Figure  4(c). Figure 4 Outward expansion and inward deformation of EDMMR: (a) Expansion deformation contour maps at different times, (b) Temperature loading mode and overall strain rate of EDMMR, (c) Nodal displacements of porous nodes at different times R en et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 5 of 14 The joint deformation exhibits distinct sectional char - affecting the contact state. The thermal expansion and acteristics. Initially, as the temperature rises (illustrated contact between the wires are interdependent and form in Figure  4(c) stage a  and  b), the displacement of the an interactive dynamic process. To gain a deeper under- joint increases linearly with time. However, as time fur- standing of this phenomenon, a post-processing func- ther elapses (illustrated in Figure  4(c) stage b  and  c), tion based on finite element contact search is utilized to the deformation is accommodated by the pore and thus extract the contact evolution process of the model during suppressed. the heating process, as illustrated in Figure 6. When utilizing the material in a high-temperature The contact density within the model varies due to environment, it is crucial to consider the deformation of differences in relative density. However, the contact the structure size and the resulting thermal stress, tak- behavior gradually strengthens, transitioning from slip ing into account the impact of stress generated by ther- to stick and from distant to near contacts. To quantita- mal expansion on the properties of EDMMR material. tively analyze the dynamic evolution of internal contact To ensure the stability of the EDMMR component dur- in EDMMR, the number of four contact pairs under a ing its usage, it is typically necessary to apply a certain single thermal load was determined using the small ball level of pre-compression. However, due to external con- algorithm. The results are presented in Figure  7. Overall, straints and the influence of its structure, the local area of the number of contact pairs (including slip and adhesive the component cannot fully expand freely when there are contacts) increases by approximately 50% with increas- changes in ambient temperature, resulting in the genera- ing temperature. In contrast, the number of non-contact tion of corresponding stress. Based on the temperature pairs (including close and distant contacts) decreases field distribution results of the EDMMR model combined accordingly. Furthermore, adhesive contact exhibits slow with the model’s displacement, the distribution of equiv- growth, while slip contact dominates and exhibits a sig- alent stress can be calculated, as depicted in Figure 5. nificant increasing trend. Upon further temperature increase, different regions of the model exhibit varying levels of stress distribution, 3 Schapery Model Derivation for EDMMR Thermal with stress levels becoming more pronounced. Previ- Expansion ous studies have indicated that the elastic deformation 3.1 Thermal Dilation of Helical Elements and Their capacity of EDMMR decreases with increasing prestress- Combinations ing [28]. Consequently, elevated temperatures enhance The thermal expansion behavior of EDMMR in high-tem - the stiffness of EDMMR, thereby strengthening its rigid perature environments is influenced by various factors, bearing performance. including material properties and pore microstructure. Therefore, accurately predicting its thermal expan - 2.4 C ontact Form Analysis and Statistical Evaluation sion coefficient is relatively challenging. The simplest The spatial position of the metal wire inside EDMMR approach is to construct a linear model based on the mix- undergoes changes when subjected to loading, thereby ing law, where the bulk thermal expansion coefficient is Figure 5 Stress Contour maps of EDMMR under different temperature environments Ren et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 6 of 14 Figure 6 Contact form change and small ball algorithm linearly related to the thermal expansion coefficients and volume ratios of its components. However, this model neglects the elastic effect of the material during the sim - plification process. While it may yield close predictions to experimental results under specific conditions, it intro - duces significant errors in most cases. In this section, we further investigate the principles of thermal expansion deformation based on helical elements. The structural discretization reveals 211 metal wires in the model, and at the extreme coordinate point of the Y-axis, 625 helical elements can be intercepted, as depicted in Figure 8. The spiral unit undergoes expansion and deforma - tion in all directions upon heating, leading to axial Figure 7 Statistics of metal wire contact quantity Figure 8 Meso-expansion principle of EDMMR model and their equivalent models R en et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 7 of 14 displacement and radial displacement. Based on material corresponding equivalent expansion model, the influence thermodynamics and spring theory, the helical element of the combination form on the thermal expansion can experiences thermal stress due to external constraints, be analyzed, as shown in Figure 8(c). which can be categorized into axial and radial compo- For the uncontacted helical unit group, the pores act as nents [29]: an accommodating mechanism, allowing for the offset of the gap between the units. As a result, only the expansion F = F sin β, of the upper-end portion is transmitted in the molding (1) F = F cos β, direction. In this case, the equivalent thermal expan- sion coefficient α along the forming direction can be where F is the axial component of the helical element described as follows: whose axial angle can be any value, F represents the ther- mal expansion force, F is the radial component, and β is αl sin θ cos β �T the axial angle. α = (l + l sin θ cos β + l)�T 0 1 Considering that the helical element does not experi- (5) l sin θ cos β ence twisting or bending during the expansion process, = α ≤ α, 2(l + l sin θ cos β + l) 0 1 the effects of torque and bending moment can be dis - regarded. By applying Castigliano’s theorem [30], the where α represents the thermal expansion coefficient of deformation of the helical element can be determined as the stainless steel wire, l denotes the original length of the follows: spiral unit, and ∆T represents the temperature change. l Meanwhile, the lower end of the uncontacted helical  ∂F F 4FD sin θ t1 t1 L �Z = · ds = ,  unit group remains unaffected by forces during expan - ∂F EA Ed cos θ sion, resulting in an equivalent stiffness given by: � (2) ∂F F 2FD cos θ cos β  t2 t2 L  �R = · ds = , 2 k = k(β ). 1 1 (6) ∂F EA Ed For the helical element group in sliding contact, where ∆Z represents the expansion deformation of the the equivalent thermal expansion coefficient can be helical unit under axial load, ∆R represents the expan - expressed as follows: sion deformation of the helical unit under radial load, D is the diameter of the helical unit, θ is the helix angle αl sin α cos β �T + αl sin α cos β �T 1 2 α = = α. of the helical unit, β is the axial angle of the helical unit, 2 (l sin α cos β +l sin α cos β )�T 1 2 F = F sin θ , F = F sin ϕ cos θ = F sin ϕ cos β cos θ , t1 t2 r (7) E is the elastic modulus of the material, and d rep- At this stage, the helical element generates both a sup- resents the wire diameter of the coil, A = πd /4 , porting force and a reaction force. The equivalent stiff - l 2π ds = D/2 cos αdθ. 0 0 ness of the helical element can be expressed as [31]: The comprehensive deformation of the helical unit along the forming direction can be obtained as follows: k(β )k(β ) 1 2 k = . sin (|β − β | ) 1 2 1 + tan (|β − β | ) k(β ) + k(β ) 1 2 1 2 �n=�Zcosβ + �R sin β. cos (|β − β | ) (3) 1 2 (8) Then, the stiffness of the helical element along the Similarly, the equivalent thermal expansion coefficient forming direction can be calculated for any axis angle: of the helical element group in the extrusion contact rela- tionship can be determined as follows: F Ed cos θ k(β) = = , �n 4D cos β(ω (θ) + D cos β sin β(ω (θ)) 1 2 αl sin α cos β �T + αl sin α cos β �T 1 2 (4) α = = α. (l sin α cos β +l sin α cos β )�T 1 2 where ω and ω represent the structural deformation 1 2 (9) parameters of the thermal expansion of the helical ele- At this stage, the combined form of the helical unit 2 2 ment, ω = sin θ, ω = cos θ. 1 2 group resembles a parallel structure, resulting in the fol- In addition to the structural parameters of the spiral lowing equivalent stiffness: element itself, the combination form between the ele- ments also influences the thermal expansion of the overall k = k(β ) + k(β ). 3 1 2 (10) model. Therefore, representative combination forms can uTh s, the elastic modulus of the structure can be be summarized based on the different relative positions expressed by its stiffness: and contact states of the helical units. By establishing the Ren et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 8 of 14 The proportion of uncontacted pairs is the largest in E = k A, i = 1, 2, 3, (11) i i EDMMR materials at low temperatures due to the pres- where E (i = 1, 2, 3) represents the elastic moduli of the ence of pores. Consequently, the uncontacted pairs have three fundamental combinations of helical unit groups. the most significant influence on the thermal expansion It can be observed that the thermal expansion of the coefficient of EDMMR. As the temperature increases, uncontacted helical unit is constrained by the presence of the thermal expansion deformation gradually intensi- gaps, limiting its transfer from forming to thermal expan- fies. The thermal expansion deformation, resulting from sion. On the other hand, the contacting helical unit directly external constraints and material expansion, affects the transfers the deformation from forming to thermal expan- gaps within EDMMR, leading to changes in its internal sion through contact, unaffected by other factors. contact state. Additionally, Figure 9 presents the thermal expansion of EDMMR samples at different temperatures 3.2 P rediction of Thermal Expansion Coefficient based on the Schapery model of finite elements and the Through the derivation of the above formula, it can be thermal expansion behavior of EDMMR. observed that the equivalent expansion coefficient and Based on the analysis of the thermal expansion perfor- elastic modulus of any helical unit group are closely mance of EDMMR, it is observed that the thermal expan- related to its axis angles β and β . In this regard, the axis sion coefficient of each component is influenced by the 1 2 angle distribution of the spiral units inside the model is volume fraction and elastic modulus of the combined calculated based on structural discrete statistics. On this form. The equivalent thermal expansion coefficient of the basis, a random distribution combination of metal wires uncontacted helical unit group is correlated with the rela- at any angle is established to capture the complex and tive density of EDMMR. Similarly, for the spiral element disordered discontinuous structure of metal wires within group in sliding and adhesive contact, the thermal expan- the EDMMR. sion coefficient is solely influenced by the thermophysi - The Schapery model [32], commonly used in woven cal properties of the EDMMR material. The macroscopic composite materials, provides a description of the influ - thermal expansion performance of EDMMR is primar- ence of each component’s elastic modulus on the overall ily influenced by its structure. Furthermore, the volume thermal expansion coefficient based on the energy prin - fraction of each component in EDMMR is directly related ciple. In the case of EDMMR, the various contact forms to the relative density of the material. exhibit different effects on the thermal expansion coef - ficient, similar to materials with different components. 4 Experimental Verification Additionally, the thermal expansion performance, vol- 4.1 Experimental Instrumentation and Principles ume fraction, and elastic modulus of EDMMR vary under EDMMR is composed of metal wires surrounded by different relative densities, resulting in different con - pores filled with a gaseous medium. The pore structure tributions to the macro thermal expansion of EDMMR. has varying degrees of impact on the material’s macro- Considering the impact of different contact forms on the scopic properties, including thermal expansion proper- thermal expansion behavior of EDMMR, it can be con- ties. The pore structure of EDMMR is characterized by cluded that: parameters such as porosity, shape, and pore size distri- bution. Typically, the internal pores of EDMMR exhibit N N α = α ϕ E ϕ E , e i i i i i (12) 1 1 where α represents the equivalent thermal expansion coefficient of the entire structure, and α , E , and ϕ denote i i i the thermal expansion coefficient, elastic modulus, and volume fraction of the respective components. By substituting the derived volume fraction, equiva- lent thermal expansion coefficient, and equivalent elastic modulus of the different combinations of helical element groups into the following formula, the Schapery model for EDMMR can be established, enabling the calculation of its equivalent thermal expansion coefficient: α ϕ E + α ϕ E + α ϕ E 1 1 1 2 2 2 3 3 3 α = . MR (13) ϕ E + ϕ E + ϕ E 1 1 2 2 3 3 Figure 9 Prediction of thermal expansion result of EDMMR R en et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 9 of 14 random distribution and irregular shapes, making it chal- Table 3 Parameters of different sample groups lenging to characterize the structure using conventional Samples serial Average Average Average methods. However, an equivalent diameter can be calcu- number height (mm) diameter (mm) relative density (%) lated using a simplified model. To investigate the pore structure of EDMMR, elec- R–1 8.00 8.00 – tron microscopy was used to observe the internal pores. M–4 8.02 8.01 29.8 Porosity and pore size distribution of the EDMMR sam- M–5 8.03 8.02 40.3 ples were measured using the automatic mercury injec- M–6 8.05 8.01 50.4 tion instrument MAC AutoPore IV 9500, as shown in Figure 10(a). The measurement is based on the Washburn equation [33], which relates the pore size to the mercury parameters presented in Table  3. The experimental pro - contact angle with the material and the applied mercury cedure involved placing different sample groups onto the pressure. sample holder and enclosing them within a heating fur- The thermal expansion experiment of EDMMR uti - nace. The ambient temperature was gradually raised to lized the DIL 402C thermal dilatometer manufactured by the desired temperature using a thermocouple as the heat NETZSCH, Germany. The experimental setup includes source, with a heating rate of 5 ℃/min [34]. Due to the a temperature-controlled furnace, a test tube sample material’s efficient heat transfer capability and the small holder, a displacement sensor, and other components, size of the samples, which ensured prolonged insulation as shown in Figure  10(b). The thermal dilatometer offers time, the temperature difference between the samples excellent linearity and resolution (0.125 μm, 0.1 K) within and the environment could be disregarded. Therefore, the a wide measurement range (−5000 μm to +5000 μm) and temperature inside the furnace was considered equiva- temperature control range (−180 ℃ to 2000 ℃). lent to the sample temperature. Additionally, high-purity During the experiment, the sample is subjected to ther- argon gas was introduced at a flow rate of 50 mL/min as mal expansion by the heating unit, causing the pusher to a protective gas to maintain uniform sample temperature move along the linear guide of the sample container. The distribution, prevent oxidation, and ensure a constant displacement sensor, connected to an optical encoder, pressure. Finally, the expansion data of the samples were directly measures the corresponding length change periodically measured and recorded. on the appropriate scale. For clarity, the sample, heat- ing unit, pusher, linear guide, and displacement sensor 4.2 Experimental Results and Discussion are color-coded in Figure  10(b) for easier visualization: The density of the wire and the pore size of the EDMMR purple, red, green, blue, and yellow, respectively. Impor- sample are directly affected by different porosity. Scan - tantly, the binding force between the push rod and the ning electron microscope (SEM) was used to observe the sample is negligible during this process, allowing for EDMMR samples with varying porosity, as shown in Fig- thermal expansion in a free state. ure 11. It can be observed that the pores surrounding the To investigate the structural stability of thermal metal wire exhibit an overall irregular geometric shape. expansion in EDMMR, EDMMR samples of the same With an increase in porosity, the arrangement of wires size were included as a control group, with sample inside the sample becomes more sparse. Figure 10 Pore structure and thermal expansion experiments of EDMMR: (a) Mercury intrusion measuring instrument, (b) Thermal dilatometer and experimental schematic Ren et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 10 of 14 Figure 11 SEM images of the EDMMR samples with different porosity levels Figure 13 Variation curve of thermal expansion of EDMMR with Temperature the pores can accommodate the inward expansion of the metal wire, thereby enhancing the structural sta- bility of the EDMMR component in high-temperature environments. The thermal expansion test was conducted on the Figure 12 Mercury intrusion experiment curves samples within a temperature range of 25 ℃ to 700 ℃. The measurement focused on the thermal expansion deformation of EDMMR during the heating process. The variation of the linear expansion coefficient in the Table 4 Experimental pore size distribution results forming direction with temperature was calculated and Sample porosity Aperture concentration Immersion recorded. The results are presented in Figure  13. Com- (%) distribution value (mm) amount −1 paring the change curve of the rubber material (used as (g·mL ) a control) with that of EDMMR, it can be observed that 30 0.142 0.380 the EDMMR exhibits better structural stability. Initially, 40 0.247 0.631 the curve shows nearly linear growth as the temperature 50 0.288 1.014 increases. However, at specific temperatures (287.5  ℃, 514.0 ℃, 615.1 ℃, and 635.7 ℃), the sample encounters phase transformation softening points. This leads to var - The equivalent pore size inside the samples with ying degrees of shrinkage due to the relaxation of internal different porosity was measured using the mercury strain, which differs from the behavior observed in other intrusion method. Figure  12 and Table  4 present the materials. In the case of entropy elastic material such as mercury intrusion curves and pore size distribution rubber, the softening shrinkage can be thermodynami- results of the EDMMR samples. The mercury intru - cally explained as a shrinkage occurring under constant sion curves exhibit symmetry around their mean val- tension as the temperature increases. The rate of thermal ues and approximate a normal distribution, which has shrinkage can be expressed as: been verified by previous studies using the plane ran - ∂L L 1 dom segmentation theory [35]. Moreover, as the poros- =−f · · , (14) ity increases across different samples, the average pore ∂T T E size also increases. By comparing the minimum aper- where f is the tension per unit area, T is the absolute ture concentration distribution value (0.142  mm) with temperature during the thermal expansion test, L is the the maximum deformation result (0.096 mm) obtained length of the sample in the load direction, L is the initial from the finite element simulation, it is evident that length of the sample in the load direction, and E is the the latter is smaller than the former. This indicates that elastic modulus. R en et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 11 of 14 The elastic component of EDMMR primarily exhibits show good agreement, indicating that these coefficients energy elasticity, which results in its temperature tol- are inherent properties of the material. Furthermore, erance and wider range of temperature stability. It can the physical expansion coefficient remains relatively maintain a stable bearing capacity, characterized by stiff - constant until a specific softening point, demonstrating ness and hardness, even in high-temperature environ- the expansion stability of the EDMMR sample at high ments. Only when the temperature exceeds its limit, its temperatures. performance will be weakened, although the reversible deformation value remains small. Furthermore, the slope 4.3 Residual Analysis of the linear phase in the EDMMR curve is influenced by Residual analysis was conducted by comparing the pre- the porosity of the sample and increases as the porosity dicted thermal expansion with the experimental results, decreases. as presented in Figure 15. The corresponding data can be Furthermore, the sample’s physical expansion coeffi - found in Table 5. cient and engineering expansion coefficient can be deter - Based on the preceding analysis, the residual points mined as follows: in the horizontal band area are evenly distributed, dem- onstrating the effectiveness of the constructed predic - 1 dL tion model in capturing the expansion characteristics of α(T ) = · , (15) L dT EDMMR. To further assess the model’s validity, the cor- relation index R can be calculated using the residual sum of squares (RSS) and the total sum of squares (TSS). �L �L 1 α = − · , (T −T ) 1 2 L L T − T 1 1 2 1 (T ) (T ) 2 1 RSS = (y − yˆ ) , i i (17) (16) i=1 where dL/dT represents the partial derivative of length with respect to temperature, ∆L is the sample elongation, ∆T is the temperature change, L is the original length of 2 TSS = (y − y) , i (18) the sample, T is the fixed reference temperature (22  ℃), i=1 and T is the maximum heating temperature. The instantaneous change rate of the linear size of the RSS EDMMR sample in the forming direction is represented R = 1 − , (19) TSS by the former, while the latter represents the average change rate of the linear length of the sample within a where, n is the number of sample observations, y is specific temperature range (T , T ). The change curves the sample observation value, y ˆ is the model predic- 1 2 of these results with temperature are shown in Fig- tion value, y represents the average value of sample ure 14(a) and (b), respectively. The physical and engineer - observations. ing expansion coefficient curves of the EDMMR samples Figure 14 Graphs depict the curves of different expansion coefficients for EDMMR Ren et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 12 of 14 Figure 15 Residual analysis of the studied EDMMR samples Table 5 Experimental results and predicted values of thermal expansion of samples at different temperatures (the predicted values in parentheses) −3 −3 −3 Temperature (℃)M–4 (10 )M–5 (10 )M–6 (10 ) 100 2.53 (2.31) 2.20 (2.00) 0.82 (1.30) 200 6.60 (6.93) 5.80 (6.10) 3.90 (4.30) 300 10.6 (10.4) 9.50 (9.10) 7.30 (6.80) 400 14.5 (14.4) 13.8 (13.3) 10.9 (10.3) 500 18.2 (18.0) 17.8 (17.2) 12.5 (11.9) (1) The internal topological structure of EDMMR was Table 6 Accuracy analysis of the model effectively reconstructed using VDT, followed by Sample serial RSS TSS R finite element thermal-solid coupling analysis. The number results showed that with the increasing tempera- −7 −4 M–4 2.66×10 2.47×10 0.9892 ture, the stress in the material became concentrated −7 −4 M–5 9.00×10 2.34×10 0.9616 at both ends and gradually increased, leading to a −6 −4 M–6 1.36×10 1.35×10 0.8994 more significant rigid support performance. Fur - thermore, EDMMR exhibited enhanced struc- tural stability compared to other porous materials, thanks to its unique pore accommodation mecha- The results are presented in Table  6, indicating the cor- nism. relation indices R for the three samples. A higher value (2) The thermal expansion behavior of EDMMR is of R indicates greater prediction accuracy and a stronger mainly governed by the interactions among the linear correlation between the observed and predicted contacting helical coils. Furthermore, the thermal variables. It can be observed that R values for all three expansion properties, volume fractions, and elas- samples are close to 1. However, as the porosity increases, tic moduli of the individual components within the irregularity of the samples also increases, resulting in EDMMR exhibit variations, leading to different higher errors. contributions to the overall macroscopic thermal expansion. The transition between gaps among 5 Conclusions uncontacted microelements and different contact This study conducted finite element simulation, theo - forms plays a crucial role in determining the ther- retical analysis, and experimental verification to investi - mal expansion coefficient of EDMMR under vary - gate the thermal expansion deformation mechanism of ing relative densities. EDMMR at high temperatures. The research methodol - (3) A modified prediction method was developed for ogy is described as follows: the thermal expansion coefficient of EDMMR, incorporating structural discretization and random distribution of contacts. This method considers the R en et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 13 of 14 [9] B Gadot, O R Martinez, S R Du Roscoat, et al. Entangled single-wire NiTi macroscopic parameters of EDMMR and allows for material: A porous metal with tunable superelastic and shape memory accurate prediction of its thermal expansion perfor- properties. Acta Materialia, 2015: 311-323 mance under different temperatures, considering [10] Y Yang, Z Y Ren, S Y Zhao, et al. One-step fabrication of thermal resistant, corrosion resistant metal rubber for oil/water separation. Colloids and various preparation and material parameters. The Surfaces A: Physicochemical and Engineering Aspects, 2019: 157-164 results demonstrate a positive correlation between [11] T Li, H B Bai, X Xue, et al. Fatigue properties of knitted-dapped metal rub- relative density and temperature in relation to the bers under high temperature environment. China Mechanical Engineering, 2019, 30(9): 9. (in Chinese) thermal expansion behavior of EDMMR. [12] Z Y Ren, W Gang, Z G Guo. Biomimetic high-intensity superhydrophobic metal rubber with anti-corrosion property for industrial oil-water separa- tion. New Journal of Chemistry, 2018, 43(4). Acknowledgements [13] Z Y Ren, D D Wu, L W Shi, et al. Research on the structural performance of Not applicable. large aspect ratio O-ring metal rubber seals based on virtual preparation. Lubrication Engineering, 2023, 48: 40-47. (in Chinese) Authors’ Contributions [14] Z N Zhang, N Yin, S Chen, et al. Tribo-informatics: Concept, architecture, ZR and XS were responsible for formal analysis, visualization, and editing; and case study. Friction, 2021, 9(3): 14. QW contributed to the investigation, conceptualization, and conducting the [15] F Q Lai, X F Hao, N N Liu, Y, et al. Creep properties of cylinder metal rubber experiments; RF and ZH provided assistance with methodology and resources. under static compression at elevated temperatures. Symmetry, 2023, All authors read and approved the final manuscript. 15(2). [16] F L Cao, H B Bai, G Q Ren, et al. Constitutive model of metal rubber Funding material based on curved cantilever beam of variable length. Journal of Supported by National Natural Science Foundation of China (Grant Nos. Mechanical Engineering, 2012, 48(24): 61. U2330202, 52175162, 51805086, 51975123), Fujian Provincial Technological [17] Y H Ma, X L Tong, B Zhu, et al. Theoretical and experimental study on Innovation Key Research and Industrialization Projects (Grant Nos. 2023XQ005, thermophysical properties of metal rubber. Journal of Physics, 2013, (4): 2024XQ010), Project of Guangdong Provincial Science and Technology Bureau 10. (in Chinese) of Jiangmen City (Grant No. 2023780200030009506). [18] H Ye, M Y Ma, J L Yu. Anomalies in mid-high-temperature linear thermal expansion coefficient of the closed-cell aluminum foam. Science Bulletin: Data availability English version, 2014: 3669–3675. The data that support the findings of this study are available on request from [19] S Chen, J Marx, A Rabiei. Experimental and computational studies on the the corresponding author, [Shi], upon reasonable request. thermal behavior and fire retardant properties of composite metal foams. International Journal of Thermal Sciences, 2016: 106–170. [20] A Takezawa, M Kobashi. Design methodology for porous composites Declarations with tunable thermal expansion produced by multi-material topology optimization and additive manufacturing. Composites Part B: Engineering, Competing Interests 2017, 131: 21-29. The authors declare no competing financial interests. [21] S Roy, A Nagel, K A Weidenmann. Anisotropic thermal expansion behav- ior of an interpenetrating metal/ceramic composite. Thermochimica Acta, 2020, 684. Received: 27 February 2023 Revised: 19 December 2024 Accepted: 23 [22] T Li, H B Bai, F L Cao. A quasi-static compression constitutive model December 2024 for knitted-dapped metal rubber considering temperature effect. Acta Aeronautica et Astronautica Sinica, 2018, 39(10). (in Chinese) [23] M J Huang, Y L Fu, X X Qiao, et al. investigation into friction and wear characteristics of 316L stainless-steel wire at high temperature. Materials, 2023, 16(1). References [24] S J Povolny, G D Seidel, C Tallon. Investigating the mechanical behavior [1] Z Y Ren, R Z Fang, X C Chen, et al. Study on anisotropic constitutive prop- of multiscale porous ultra-high temperature ceramics using a quasi-static erties of metal rubber based on virtual fabrication technology. Journal of material point method. Mechanics of Materials, 2021, 160(3): 103976. Mechanical Engineering, 2021, 57(24): 211-222. [25] J Y Kong, H Y Zou, K Zeng, et al. Investigation of optimization methods [2] L W Shi, Z Y Ren, Z H Huang, et al. Research on trajectory optimization of for metal foam with two-dimensional porosity gradient in shell-and-tube multilateral thin metal rubber automatic laying based on virtual fabrica- latent heat storage. Journal of Energy Storage, 2023, 63: 107004. tion technology. Advanced Engineering Materials, 2021, 24(2). [26] Ren Z, Shen L, Huang Z, et al. Study on multi-point random contact [3] C H Zhou, Z Y Ren, Y X Lin, et al. Hysteresis dynamic model of metal rub- characteristics of metal rubber spiral mesh structure. IEEE Access, 2019, 7: ber based on higher-order nonlinear friction (HNF). Mechanical Systems 132694-132710. and Signal Processing, 2023: 189. [27] K C Mills. Recommended values of thermophysical properties for selected [4] P Yang, H B Bai, X Xue, et al. Vibration reliability characterization and commercial alloys. Woodhead Publishing, 2002. damping capability of annular periodic metal rubber in the non-molding [28] C Kartik, R Jem, C Elizabeth. Mechanical behaviour of tangled metal wire direction. Mechanical Systems and Signal Processing, 2019, 132: 622-639. devices. Mechanical Systems and Signal Processing, 2019, 118: 13-29. [5] X Xue, Y H Wei, Fang W, et al. Fabrication technology and shear failure [29] Q W Wang, Z Y Ren, L W Shi, et al. Stiffness constitutive characteristics of behaviours of elastic–porous sandwich structure with entangled metallic elastic disordered microporous metal rubber considering temperature wire mesh. Thin-Walled Structures. 2022, 170: 108599. effects. Advanced Engineering Materials, 2023, 25(23). [6] Y Zhu, Y W Wu, H B Bai, et al. Research on vibration reduction design of [30] W P Hu, Q C Meng. Understand the application of Karnaugh’s theorem foundation with entangled metallic wire material under high tempera- from the principle of virtual work. Mechanics in Engineering, 2019, 41(4): 4. ture. Shock and Vibration, 2019: 1-16. [31] B Zhu, Y H Ma, D Y Zhang, et al. A constitutive model of metal rubber [7] H Y Li, Z Y Ren, X L Su, et al. Study on the fretting wear evolution model of based on hysteresis property. Journal of Physics, 2012, 61(7): 8. wires with curvature inside metal rubber. Tribology Letters, 2023, 71(1). [32] R A Schapery. Thermal expansion coefficients of composite materials [8] W Zhang, X Xue, H B Bai. Mechanical and electrical properties of Cu-steel based on energy principles. Journal of Composite Materials, 1968, 2(3): bimetallic porous composite with a double-helix entangled structure. 380-404. Compos. Struct., 2021, 255: 112886 [33] E W Washburn. The dynamics of capillary flow. Physical Review Journals Archive, 1921, 17(3): 273-283. Ren et al. Chinese Journal of Mechanical Engineering (2025) 38:61 Page 14 of 14 [34] J Jin, W Bao, P Zhang, et al. Creep property of TMCP high-strength steel Q690CFD at elevated temperatures. Journal of Materials in Civil Engineer- ing, 2020, 32(2): 4019361-4019364. [35] Y G Guo, Y H Xia, Z B Chen, et al. Study on pore size distribution charac- teristics of metal rubber filter materials. Journal of Filtration & Separation, 2008. (in Chinese) Zhiying Ren born in 1980, is currently a professor and doctoral supervisor at School of Mechanical Engineering and Automation, Fuzhou University, China. She received her PhD degree from Fuzhou University, China, in 2015. Her research interests include the research of metal rubber materials and equipment vibration and noise reduc- tion technology. Qinwei Wang born in 1999, is currently a master candidate at Institute of Metal Rubber and Vibration Noise, School of Mechanical Engi- neering and Automation, Fuzhou University, China. Rongzheng Fang born in 1997, is currently a master candidate at Institute of Metal Rubber and Vibration Noise, School of Mechanical Engi- neering and Automation, Fuzhou University, China. Zihao Huang born in 1998, is currently a master candidate at Insti- tute of Metal Rubber and Vibration Noise, School of Mechanical Engi- neering and Automation, Fuzhou University, China. Xianjie Shi born in 1985, is currently a senior engineer at Institute of Systems Engineering, China Academy of Engineering Physics, China. He received his PhD degree in mechanical design and theory from Harbin Engineering University, China, in 2014.

Journal

Chinese Journal of Mechanical Engineering – Springer Journals

Published: May 12, 2025

Keywords: Elastic disordered microporous metal rubber (EDMMR); Virtual manufacturing technology (VMT); Elevated temperature; Coefficient of thermal expansion (CTE)

There are no references for this article.