Heat transfer analysis and thermo-mechanical property of M-shape metal-rubber at high-temperature

Particular attention is given to the M-shape metal-rubber (MMR), which is a porous material with cushioning characteristics and excellent heat resistance coated in pipelines. The heat transfer characteristics and thermo-mechanical property of MMR has not been clear due to its complicated micro-porous structures and the coupling effect between structure and performance. In this work, the influence of different parameters (emissivity, density and gap distance) on the equivalent thermal conductivity of MMR were considered at the mesoscopic model level. In addition, thermal stress at different temperatures was used as predefined fields for thermal-mechanical coupling to analyze the transformations of performance. The results indicate that surface emissivity and cavity emissivity during heat transfer exhibit an inverse relationship with MMR thermal conductivity. Furthermore, it is observed that natural convection has the most significant impact on thermal conductivity. And, the thermal conductivity is negative correlation with density and gap distances. Under cyclic loading, stress concentration and plastic strain occur in MMR, and residual thermal stress is generated. The secant stiffness exhibits a positive correlation with temperature, while the loss factor demonstrates a negative correlation, primarily due to thermal expansion.


Introduction
Pipeline systems have been widely used in the fields of engine, power system of submarines and various warships [1].To mitigate the effects of vibration and high temperature, Metal-rubber (MR) damper cladded for pipe system is an alternative to achieve this objective [2].MR materials have gained widespread usage due to properties, including high elasticity, rigid damping, high temperature resistance and corrosion resistance [3].MMR, as a topology optimization structure of MR, offers the advantage of designable stiffness.Its notable attributes, including large stroke cushioning capabilities under impact loads and thermal protection properties in high-temperature environments, making it crucial in pipeline coating structures.Therefore, it is vital to explore the heat transfer characteristics and thermo-mechanical property in case of MMR.
Predicting the thermal conductivity of MR materials is challenging due to internal porous structure with random interlocked wire mesh.Various methods can be employed to assess its heat transfer performance based on the heat transfer mode of porous materials.These methods include the thermal resistance network method [4], finite element method [5], transient plane source method [6], and so on.
This work focuses on conducting heat transfer simulations of the MMR coated pipeline system.The equivalent thermal conductivity (ETC) model is utilized to predict the thermal conductivity of the system and compare with Macroscopic model.The analysis aims to examine the variations in thermal conductivity under different heat transfer conditions.Additionally, the thermo-mechanical coupling simulation is carried out as a predefined field under thermal stress at different temperatures, and the mechanical properties at different temperatures were analyzed.

The heat transfer mechanism and model of MMR
From the perspective of heat transfer, there is a certain porosity in the MR structure.The air between the holes combines with the metal wire spiral coil to form a sandwich structure for heat transfer within the pipeline.All heat sources come from pipes subjected to high temperatures, and when they come into contact with MMR, various heat transfer conditions such as conduction, radiation and convection occur.Fig. 1(a) exhibits the entire MMR protection is subjected to macroscopic heat transfer, the cavity radiation generated by itself, the natural convection and surface radiation of the down panel to outside.As illustrated in Fig. 1(b), the three-dimensional model was created using by generate a spiral structure, and then imported into Abaqus for compression simulation.The material utilized in the simulation is 316L stainless steel, and the mesh type employed was C3D8R.Table 1 and Table 2 provided the properties of 316L stainless steel and air at different temperatures, respectively.Upon completion of the compression simulation, the MMR model was imported into the heat transfer analysis step using the post-processing module.The element type was modified to DC3D8.The model was positioned in air to simulate natural convection.It's assumed that the panels in question are extremely thin.The temperature boundary condition was applied to the up panel, while radiation and convection boundary conditions were applied to the down panel.Transient heat transfer processes were adopted to obtain the heat flux.According to the simulation results, the heat flow and temperature of the MMR surface were obtained, and the effective conductivity of the thermal conduction process was calculated according to the temperature.The equivalent thermal conductivity can be calculated as: where q is the heat flux generated for all conditions.T  is the temperature subtraction between the up and down panels.c h is the distance of up and down panel.This process can be described by threedimension transient heat transfer equation. (2)

Heat transfer analysis
By applying a constant temperature boundary of 500°C to the up panel, the thermal conductivity of the mesoscopic model was calculated using the aforementioned equation.A macroscopic model with identical dimensions as that of the meso-model was created by incorporating the obtained thermal conductivity.As depicted in Fig. 2(a) and (b), both models exhibit similar temperature gradients and variations in heat flux.The meso-model demonstrates higher temperature consistency and longer transfer path.Fig. 2(c) illustrates the changes in temperature gradient at equidistant points along the thickness direction.As depicted in Fig. 2(a) and (b), both models exhibit similar temperature gradients and variations in heat flux.The meso-model demonstrates higher temperature consistency and a longer heat transfer path.However, as we move away from the temperature boundary, the temperature of the meso-model is higher than that of the macro-model, indicating a positive correlation.The number of different spiral wire circles was changed to obtain meso-models of different MMR densities.In this work, three kinds of MMR with different relative densities were prepared, which were 1.5g/mm 3 , 2g/mm 3 and 2.5g/mm 3 .The influence of density on results is in Fig. 4(a).As the density increases, its equivalent thermal conductivity decreases.At 100℃, when the density increases from 1.5g/mm 3 to 2.5g/mm 3 , its thermal conductivity decreases by 58% and 500°C by 52%.The reason is that MMR of different densities, the number of spiral turns is different.According to the Thermo-electric Analogy, an increase in the number of turns compresses the internal porosity of the material, leading to a decrease in the thermal resistance of the air.However, due to the parallel relationship, the overall thermal resistance value increases, resulting in a reduction in thermal conductivity.
To prevent friction on the outer wall of the pipeline and minimize roughness, the coating should maintain a certain non-contact distance from the pipeline's outer wall.In non-contact heat transfer, an air gap exists between the MMR and the thermal boundary (pipe panel).Fig. 4(b) demonstrates that the thermal conductivity decreases significantly as the distance between the air layers increases.This occurs because the presence of the air layer causes the piping system to connect the air thermal resistance in series, increasing the overall thermal resistance and lowering the temperature of the down panel.

Thermo-mechanical coupling analysis of MMR
In this section, quasi-static loading and unloading simulations of MMR at different temperatures were conducted.Building upon the results obtained from the heat transfer simulations at 100°C, 300°C, and 500°C in the previous section, the Dynamics-Explicit module was employed to incorporate a predefined temperature field.The load module applied a 1mm loading and unloading displacement to the MMR, observing the thermal stress under thermo-mechanical coupling.The results depicted in Fig. 5(a) demonstrate that an increase in temperature intensifies local extrusion and stress concentration, leading to the occurrence of irreversible plastic strain.Fig. 5(b) presents the changes in equivalent stress at nodes experiencing stress concentration at different temperatures.It was observed that the stress at these nodes generally exhibits a positive correlation with compression, with its growth phase subsequently slowing down and even experiencing a slight decline.During the unloading phase, it is observed that the MMR gradually returns to its original state as the displacement returns to 0. However, due to irreversible plastic deformation, residual stress remains even after the load is removed.This residual stress is predominantly concentrated in areas where the thermal stress is higher.As the load diminishes, the structure undergoes slight warping, causing the metal wires on both sides to compress inward.The load-displacement curves for the entire process are presented in Fig. 6.It can be noted that an increase in temperature enhances the secant stiffness of the MMR and strengthens its load-bearing capacity.The curve during the unloading phase lags behind the loading phase curve, indicating the occurrence of frictional energy dissipation within the MMR.With higher temperatures, the rebound ability of the MMR increases.However, the expansion of the wires compresses the air pores, resulting in reduced sliding space and a decrease in the loss factor η.
In the thermo-mechanical coupling simulation, a quasi-static loading and unloading curve was employed for the MMR.The quasi-static approach allows for a certain velocity and large displacement while disregarding the inertial forces in the process.It is suitable for simulating slow loading scenarios.However, it is important to verify if this approach closely approximates the actual working conditions.As a general rule, the kinetic energy of the deforming material should not exceed a small fraction (typically 5% to 10%) of its internal energy throughout most of the process.To assess the applicability of the quasi-static approach, the internal energy (ALLIE) and kinetic energy (ALLKE) were derived from the time history curve of the simulation results, as shown in Fig. 6(d).Qualitatively, ALLKE represents the kinetic energy calibration.The values of kinetic energy cannot be excessively large using static method when solving dynamic problems.From the results, it can be observed that the maximum point of the ratio (ALLKE/ALLIE) occurs at the beginning of the loading process, reaching 2.95%.Subsequently, it gradually decays, and the entire process levels off.Thus, based on this analysis, the process can be considered quasi-static.

Conclusion
In this work, the heat transfer analysis of MMR was performed using the finite element method, as well as the calculation of effective thermal conductivity.The thermo-mechanical properties of MMR were obtained based on numerical method.The main conclusions can be summarized as follows: (1) Emissivity has a significant influence on the thermal conductivity of MMR.The thermal conductivity is positively correlated with surface emissivity and negatively correlated with cavity emissivity.Natural convection has the greatest effect on the overall thermal conductivity in all transfer conditions.The thermal conductivity is in a negative correlation to the density and gap distances of MMR.
(2) Under the influence of thermo-mechanical coupling, stress concentration and plastic strain occur in MMR, and residual thermal stress is generated.The stress tends to stabilize as the displacement increases.The secant stiffness of MMR in the whole loading-unloading process is positively correlated with temperature and the loss factor is negatively correlated due to thermal expansion, which shows the structural stability and heat resistance of MMR.

Figure 1 .
Figure 1.(a)The heat transfer mechanism of MMR (b) Heat transfer model of MMR

Figure 2 .
Figure 2. The difference of Mesoscopic model and Macroscopic model at 500 ℃ (a) Temperature gradient (b)Heat flux (c) Temperature variation at three equidistant nodes in thickness directionDifferent emissivity values were applied to the meso-model to assess their impact on thermal conductivity.The variations in thermal conductivity were then compared between surface emissivity and cavity emissivity, as illustrated in Fig.3.Remarkably, both emissivity values exhibited distinct trends in influencing the thermal conductivity of MMR.This intriguing disparity arises due to the fact that surface emissivity facilitates the radiation of heat to the surrounding environment, whereas cavity emissivity stores heat, leading to a reduction in heat circulation.Fig.3(b) presents a comparison of the influence of various thermal conditions on the thermal conductivity of MMR.The study takes into account the effects of conduction, radiation, and convection on thermal conductivity.The results reveal that when considering only conduction, MMR exhibits the lowest thermal conductivity.However, when natural convection is taken into account, the

Figure 3 .
Figure 3. (a) Effect of emissivity on heat transfer (b) Thermal conductivity of MMR under different conditions

Figure 4 .
Figure 4. (a) The effect of density on heat transfer (b) The effects of non-contact distance on heat transfer.

Figure 5 .
Figure 5. (a) Stress changes at different temperatures (b) Stress concentration points at different temperatures

Figure 6 .
Figure 6.(a) The hysteresis process of MMR (b) Load-displacement curves at different temperatures (c) Parameter characterization (d) Quasi-static error curve.