Determination of Elastic Modulus, Stress Relaxation Time and Thermal Softening Index in ZWT Constitutive Model for Reinforced Al/PTFE

Al/PTFE has the advantages of high impact-responsive energy release, appropriate sensitivity, a fast energy release rate, and high energy density, and it is increasingly widely being used in the field of ammunition. In this paper, based on the traditional formula Al/PTFE (26.5%/73.5%), the reinforced Al/PTFE active materials are prepared by the process of cold pressing, sintering, and rapid cooling. Quasi static and dynamic compression experiments were carried out under different compression pressures (200~800 MPa), strain rates (0.002 s−1, 0.02 s−1, 1400~3300 s−1), and temperatures (23 °C, −20 °C, −30 °C, −40 °C). The effects of pressure, strain rate, and temperature on the quasi-static and dynamic compression properties of Al/PTFE materials are analyzed. The results show that the reinforced Al/PTFE specimens show a significant correlation between temperature and strain rate. Based on the classical Zhu–Wang–Tang (ZWT) constitutive model, the ZWT constitutive model parameters of the reinforced Al/PTFE active materials under different pressing pressures at room temperature and the ZWT constitutive model parameters of the reinforced Al/PTFE active materials at low temperature are obtained by fitting, respectively. The accuracy of the constitutive model parameters (elastic modulus, stress relaxation time, and thermal softening index) is verified. In this paper, a constitutive model considering both temperature and strain rate effects is established in order to provide reference for the study of mechanical properties of active materials.


Introduction
Al/PTFE (aluminum/polytetrafluoroethylene) active materials, as energetic materials with a strong deflagration reaction and strong energy release under impact load, are widely used in ammunitions such as shaped charge liners and warhead shells due to their advantages such as appropriate sensitivity, high energy density and large amount of gas generated by reaction [1]. The strength of traditional Al/PTFE active materials under dynamic loading is only tens of MPa, which restricts the promotion of materials under the demand of high strain rate. In order to greatly improve the dynamic compressive strength of Al/PTFE active materials, the reinforced Al/PTFE are prepared using a combination of cold pressing, sintering, and rapid cooling. Traditional Al/PTFE shows obvious strain hardening and strain-rate strengthening effects in compression experiments, with strain rates ranging from 10 −3 to 10 4 [2][3][4]. It is necessary to carry out research on the mechanical properties and constitutive equations of reinforced Al/PTFE prepared by improved process.
Many scholars have constructed Johnson-Cook constitutive equations for Al/PTFE reactive materials based on quasi-static and dynamic compression test data. Xu [5] obtained the compression constitutive equation of Al/PTFE by combining a quasi-static experiment, a dynamic experiment, and microscopic analysis. It was measured that the tensile strength of the traditional formula Al/PTFE (mass ratio 26.5/73.5) active material was 24.9 MPa, shown that this constitutive model is applicable not only to the polymer itself, but also to the polymer matrix composites [21][22][23][24]. Lai et al. [25] introduced the damage function into the ZWT model to describe the constitutive relationship between ultra-high-performance cement-based composites. Wang et al. [26] modified the ZWT constitutive model by introducing strain rate and temperature effects in order to simulate the dynamic behavior of a polymethyl methacrylate (PMMA) aircraft windshield under bird impact. Xu et al. [27] studied the mechanical behavior of liquid nitrile rubber-modified epoxy resin and chose the standard ZWT nonlinear viscoelastic model to predict the elastic behavior of LNBR/epoxy composites under wide ranges of strain. Meanwhile, the authors simulated the mechanical behavior of LNBR/epoxy composites with the model parameters obtained from the experiments. Luo et al. [28] used quasi-static and dynamic compression stress-strain curves to fit the parameters of the Zhu-Wang-Tang (ZWT) constitutive equation at different temperatures. A subroutine for the ZWT constitutive model was developed in ABAQUS, and numerical simulations of split Hopkinson pressure bar tests were performed. Luo et al. [29] studied the dynamic mechanical properties and constitutive model of shale with different bedding under triaxial impact testing and established an improved dynamic constitutive relation of shale based on the ZWT model which combined with the damage theory and simplified the low-frequency term. Dong et al. [30] constructed a dynamic constitutive model of tensile and compressive damage on the basis of the ZWT and statistical damage models; the results show that the constructed dynamic constitutive model of tensile and compressive damage could considerably simulate the tensile and compressive stress-strain relations and failure features of sandstones well. Dar et al. [31] established a ZWT nonlinear viscoelastic constitutive model coupled with temperature and strain rate effects and realized the embedding of the ZWT constitutive model by establishing a user-defined material subprogram in the finite element solver LS-DYNA.
The constitutive model of the Al/PTFE active materials under the influence of low temperature and pressing pressure is still not perfect. In this paper, the dynamic compression strength of Al/PTFE active materials is greatly improved by using the preparation process of cold pressing, sintering, and rapid cooling. Static compression experiments and dynamic compression experiments at different temperatures and different strain rates are carried out on reinforced Al/PTFE specimens prepared under different pressing pressures. The static/dynamic compression mechanical behavior of the material is analyzed, and a ZWT constitutive model considering temperature is established by adding a temperature-related term to the classical ZWT constitutive model and by assuming that the parameters of the constitutive model are temperature dependent in order to describe the mechanical behavior of reinforced Al/PTFE materials at different temperatures and different strain rates. Based on the experimental data, modified ZWT constitutive model parameters suitable for reinforced Al/PTFE active materials are obtained by fitting.

Preparation of Reinforced Al/PTFE
In this paper, based on the traditional formula Al/PTFE (26.5%/73.5%), a reinforced Al/PTFE active material is prepared through improvements to the sintering process and using the rapid cooling method. The ratio and preparation technology of traditional formula Al/PTFE is proposed by Vasant [32].
Al powder is an Al particle produced by Beijing Xinyuan Technology Co., Ltd., and PTFE powder is a PTFE 7A particle produced by DuPont. An electronic balance was used to weigh the Al powder and PTFE according to the required ratio. Due to its small particle size, PTFE easily absorbs moisture and forms white clumps during storage. In order to remove the moisture in the raw material, a vacuum drying oven was used to treat PTFE. The temperature of the vacuum drying oven was set to 60 • C, the vacuum degree was 0.08 MPa, and the duration was 10 h. PTFE and Al were mixed evenly according to the proportion of 26.5%:73.5% of the mass ratio of Al: PTFE, and the mixing time was 10 min. The mixed powder was added into the self-made pressing mould. In order to avoid stratification, the mixed powder was spread in the pressing mould as far as possible by shaking the mould during the filling process. YLJ-50 was used to press the mixed powder. During the pressing process, the pressing rate was about 20 MPa/min, and the pressure was held for 5 min at the highest pressing pressure. Afterward, it was slowly unloaded in order to reduce the residual stress in the specimen, and the produced blank was obtained by demolding. The preformed billet was placed on a crucible sprinkled with quartz sand and was then pushed into the middle of the tubular Nabertherm sintering furnace with a more uniform temperature. The sintering atmosphere was N 2 . During the sintering process, the temperature ranges of 25-328 • C and 328-375 • C were the uniform heating stage, and the heating durations were 3 h 40 min and 50 min, respectively. The temperature was maintained for 30 min at 375 • C. After the insulation, the temperature was lowered to 322 • C at a constant rate within 1 h, and the temperature was maintained for 30 min. Within 1 h, the temperature had cooled to 310 • C at a constant rate, and the temperature further cooled to 0 • C after 30 min. The static compression experiments and dynamic compression experiments at different temperatures and different strain rates were carried out on the reinforced Al/PTFE specimens prepared under different pressing pressures.

Quasi-Static Compression Experiment
The quasi-static compression experiment system consisted of a loading system, a measuring system, and a temperature control system. The loading system and measuring system were carried by the WDW 100GD universal testing machine. The temperature control system was undertaken within a high-and low-temperature test chamber, which was used for temperature control in the low-temperature experiment. The liquid nitrogen tank was used to cool the Al/PTFE specimen, and the minimum temperature could reach −150 • C in the low-temperature experiment. By adjusting the liquid nitrogen flow rate and the temperature control device of the high-and low-temperature test chamber, the quasi-static compression experiment could be carried out under the required temperature conditions.
The size of the reinforced Al/PTFE specimen used in this paper is 14 ± 0.1 mm in diameter and 5 ± 0.1 mm in length at room temperature (23 • C). The quasi-static compression tests (No.1-No.3), with a strain rate of 0.002 s −1 , were carried out on reinforced Al/PTFE specimens with pressing pressures of 400 MPa, 600 MPa, and 800 MPa. The quasistatic compression tests of No.4-No.11 reinforced the Al/PTFE specimens with 200 MPa pressure were carried out at 23 • C, −20 • C, −30 • C, and −40 • C, respectively, with strain rates of 0.002 s −1 and 0.02 s −1 .

Dynamic Compression Experiment
The separation Hopkinson pressure bar (SHPB) device designed by Key Laboratory of Transient Physical Mechanics and Energy Conversion Materials of Liaoning Province was used for the dynamic loading experiment. As shown in Figure 1, the SHPB system consisted of a loading system, a velocity measurement system, a pressure bar system, a data acquisition system, and a low-temperature experimental system.  The impact velocity could be adjusted by changing the gas pressure in the gas chamber or the depth of the impact bar in the launch tube. Velocity was measured by laser and detonation velocity meter at the nozzle of the launch tube. The pressure bar system included an incident bar, transmission bar, absorption bar, and buffer device at the end. Based on the resistance strain method, the electric signals generated by the strain gauges on the incident bar and the transmission bar were collected by the ultra-dynamic strain gauge, and the corresponding stress-strain data are obtained by software processing. The deformation process of the reinforced Al/PTFE specimen during dynamic loading was recorded by PCO.Dimax HS4 high-speed camera. The impact velocity could be adjusted by changing the gas pressure in the gas chamber or the depth of the impact bar in the launch tube. Velocity was measured by laser and detonation velocity meter at the nozzle of the launch tube. The pressure bar system included an incident bar, transmission bar, absorption bar, and buffer device at the end. Based on the resistance strain method, the electric signals generated by the strain gauges on the incident bar and the transmission bar were collected by the ultra-dynamic strain gauge, and the corresponding stress-strain data are obtained by software processing. The deformation process of the reinforced Al/PTFE specimen during dynamic loading was recorded by PCO.Dimax HS4 high-speed camera.
Polyurethane foam was used as a thermal insulation material for the low-temperature loading device, and a spirally wound copper tube was inserted into polyurethane foam in order to form a cooling chamber for low-temperature experiments. The temperature was controlled by adjusting the flow rate of liquid nitrogen in the copper tube.
The stress σ, strain ε, and strain rate ε' of the specimen in the Hopkinson compression bar experiment are calculated based on the two-wave method. The calculation formula is as follows [33]: where A and A 0 are the cross-section of bars and specimen, respectively; E is elastic modulus of bars; c 0 is elastic wave velocity of bars; l 0 is the length of specimen; ε r and ε t are the strains caused by reflected and transmitted waves, respectively. Assuming the tested material was incompressible, the true stress σ T and true strain ε T of the material can be expressed as

Basic Parameters of Dynamic Compression Experiment
The reinforced Al/PTFE specimens with pressing pressures of 200 MPa, 400 MPa, 600 MPa, and 800 MPa were selected for dynamic compression experiments under three different strain rates, and the dynamic mechanical properties of the specimens under different loading pressures were analyzed. Table 1 shows the basic experimental parameters of the specimens prepared under different pressing pressures at room temperature.

Results of Quasi-Static Compression Experiment
In order to characterize the quasi-static compressive properties of the reinforced Al/PTFE specimens, using the experiment of 200 MPa of pressing pressure at room temperature and a strain rate 0.002 s −1 as an example, the tangent slope of the stress-strain curve in the elastic stage was defined as the elastic modulus E, and the tangent slope of the stress-strain curve in the yield stage was defined as the hardening modulus Et. The stress value at the intersection of the two tangent lines was defined as the yield strength σ s [5]. The schematic diagram of quasi-static compression performance parameters is shown in Figure 2.

Results of Quasi-Static Compression Experiment
In order to characterize the quasi-static compressive properties of the rei Al/PTFE specimens, using the experiment of 200 MPa of pressing pressure at roo perature and a strain rate 0.002 s −1 as an example, the tangent slope of the stress curve in the elastic stage was defined as the elastic modulus E, and the tangent s the stress-strain curve in the yield stage was defined as the hardening modulus stress value at the intersection of the two tangent lines was defined as the yield s σs [5]. The schematic diagram of quasi-static compression performance param shown in Figure 2. At room temperature, the stress-strain curves of the reinforced Al/PTFE spe under different pressing pressures at 0.002 s −1 strain rate are shown in Figure 3. At room temperature, the stress-strain curves of the reinforced Al/PTFE specimens under different pressing pressures at 0.002 s −1 strain rate are shown in Figure 3.  It can be seen from Figure 3 that the pressing pressure had little effect on the modulus and yield strength of the reinforced Al/PTFE specimen. The hardening m of the specimen prepared under 200 MPa pressing pressure was small, because the more pores in the specimen under low pressing pressure. When the material ente yield stage, the internal pores were crushed, and the specimen suddenly lost its capacity. When the pressing pressure exceeded 400 MPa, the Al particles inside th imen and the PTFE matrix were closely combined, and the stress-strain curves high degree of similarity. Figure 4 shows the quasi-static compression stress-strain curves of specimen It can be seen from Figure 3 that the pressing pressure had little effect on the elastic modulus and yield strength of the reinforced Al/PTFE specimen. The hardening modulus of the specimen prepared under 200 MPa pressing pressure was small, because there were more pores in the specimen under low pressing pressure. When the material entered the yield stage, the internal pores were crushed, and the specimen suddenly lost its bearing capacity. When the pressing pressure exceeded 400 MPa, the Al particles inside the specimen and the PTFE matrix were closely combined, and the stress-strain curves show a high degree of similarity. Figure 4 shows the quasi-static compression stress-strain curves of specimens under 200 MPa pressing pressure at different temperatures. The stress-strain curves of the reinforced Al/PTFE specimens show obvious temperature and strain rate correlation. As shown in Figure 4a, at the same strain rate, the elastic modulus, hardening modulus, and yield stress of the specimens all increased with the decrease of temperature. Under different temperature conditions, the material exhibited similar elastic behavior at the initial stage of deformation, and the elastic modulus and yield stress were not much different. However, when the strain exceeded 0.1, the hardening modulus increased significantly when the material entered the yield stage. The hardening modulus was 44 MPa at room temperature and 292 MPa when the temperature dropped to −40 • C. The decrease in temperature lead to the hardening behavior of the material, the decrease of toughness, and the decrease of the difference between the elastic modulus and the hardening modulus of the specimen. The viscoelastic property of the reinforced Al/PTFE material was weakened, and its deformation behavior is closer to that of the linear elastic material. By comparing Figure 4a with Figure 4b, when the temperature was the same, the elastic modulus, hardening modulus, and yield stress of the specimen all increased with the increase of strain rate, which has an obvious strain rate correlation.  It can be seen from the compressive stress-strain curves of the reinforced Al/PTFE specimens under different experimental conditions that the stress-strain curves can be roughly divided into the elasticity, yield, and densification stages. The material is a typical ductile material, and the yield point is not obvious, which is similar to the experimental results of Cai et al. [34]. Figure 5 shows the stress-strain curves of specimens under the same pressing pressure and different strain rates at room temperature. It can be seen that under the same pressing pressure, the strain rate had a significant effect on the dynamic mechanical properties of the specimen, while the change of pressing pressure under the same strain rate had little effect on the dynamic mechanical properties of the specimen. With the increase of pressing pressure, the microstructure of the specimen became closer, but shear cracks were more likely to appear inside the specimen, and the residual stress was easier to release during the sintering process, which cannot guarantee the structural integrity of the specimen. It can be seen from the compressive stress-strain curves of the reinforced Al/PTFE specimens under different experimental conditions that the stress-strain curves can be roughly divided into the elasticity, yield, and densification stages. The material is a typical ductile material, and the yield point is not obvious, which is similar to the experimental results of Cai et al. [34]. Figure 5 shows the stress-strain curves of specimens under the same pressing pressure and different strain rates at room temperature. It can be seen that under the same pressing pressure, the strain rate had a significant effect on the dynamic mechanical properties of the specimen, while the change of pressing pressure under the same strain rate had little effect on the dynamic mechanical properties of the specimen. With the increase of pressing pressure, the microstructure of the specimen became closer, but shear cracks were more likely to appear inside the specimen, and the residual stress was easier to release during the sintering process, which cannot guarantee the structural integrity of the specimen.

Effect of Strain Rate on Dynamic Compression
Properties of Specimens Figure 6 shows the stress-strain curves of reinforced Al/PTFE specimens under different strain rates at room temperature. At room temperature, the stress-strain curves at different strain rates show four stages: at the initial stage of loading, the stress increases rapidly with the increase of strain. The stress-strain curve at this stage is generally linear, and the elastic modulus is much larger than that of the specimen under quasi-static loading. This is due to the strain rate strengthening effect caused by the rapid increase of stress in the specimen, stress which has not reached the stress balance in the initial stage of stress wave loading. The load is primarily borne by the PTFE matrix in the initial stage. When the stress continues to increase and reaches yield stress at the matching strain rate, the specimen material yields, and the yield strength increases with the increase in strain rate. After the yield stage, the stress continues to increase with the increase of strain, and the material enters the strengthening stage. At this stage, the deformation is large, the PTFE matrix deforms sharply, and the Al particles squeeze and contact each other in order to form a force chain so that the specimen can bear higher stress. When the load exceeds the stress that the specimen can bear, the microcracks inside the specimen converge to form macro cracks, and the specimen fails.
At room temperature, the flow stress of the reinforced Al/PTFE material loaded with different strain rates increases with the increase of strain, a process which has an obvious strain-hardening effect. The dynamic yield strength of the materials increases with the increase of strain rate, and the material has an obvious strain rate-strengthening effect. The maximum stress of the stress-strain curve at each strain rate is defined as the peak stress, and the strain corresponding to the peak stress is defined as the peak strain. With the increase of the strain rate, the peak stress and peak strain of the dynamic compression of the specimen also increase monotonically. When the strain rate reaches 1400 s −1 , the  Figure 6 shows the stress-strain curves of reinforced Al/PTFE specimens under different strain rates at room temperature. At room temperature, the stress-strain curves at different strain rates show four stages: at the initial stage of loading, the stress increases rapidly with the increase of strain. The stress-strain curve at this stage is generally linear, and the elastic modulus is much larger than that of the specimen under quasi-static loading. This is due to the strain rate strengthening effect caused by the rapid increase of stress in the specimen, stress which has not reached the stress balance in the initial stage of stress wave loading. The load is primarily borne by the PTFE matrix in the initial stage. When the stress continues to increase and reaches yield stress at the matching strain rate, the specimen material yields, and the yield strength increases with the increase in strain rate. After the yield stage, the stress continues to increase with the increase of strain, and the material enters the strengthening stage. At this stage, the deformation is large, the PTFE matrix deforms sharply, and the Al particles squeeze and contact each other in order to form a force chain so that the specimen can bear higher stress. When the load exceeds the stress that the specimen can bear, the microcracks inside the specimen converge to form macro cracks, and the specimen fails.

Effect of Strain Rate on Dynamic Compression Properties of Specimens
At room temperature, the flow stress of the reinforced Al/PTFE material loaded with different strain rates increases with the increase of strain, a process which has an obvious strain-hardening effect. The dynamic yield strength of the materials increases with the increase of strain rate, and the material has an obvious strain rate-strengthening effect. The maximum stress of the stress-strain curve at each strain rate is defined as the peak stress, and the strain corresponding to the peak stress is defined as the peak strain. With the increase of the strain rate, the peak stress and peak strain of the dynamic compression of the specimen also increase monotonically. When the strain rate reaches 1400 s −1 , the maximum true stress of the reinforced Al/PTFE material is 54 MPa, which is 1.5 times that of the traditional formula Al/PTFE [35]. 2022, 14, x 10 of 24 maximum true stress of the reinforced Al/PTFE material is 54 MPa, which is 1.5 times that of the traditional formula Al/PTFE [35].  Figure 7 shows the specimens after dynamic compression at different strain rates at 20 °C. With the increase of strain rate, the radial size of the specimen increased gradually. When the strain rate is low, the size of the specimen does not change much after the dynamic compression test, while the end face of the specimen increases uniformly with the multiple loading of the stress wave, and the end face is relatively flat. When the strain rate is high, the energy carried by the stress wave is large. Before the specimen reaches the stress equilibrium state, it exceeds the deformation response limit of Al/PTFE as a viscoelastic material, resulting in local stress concentration. As a result, the center of the specimen is rapidly compressed, forming a pit. When the strain rate is 2800 s −1 , the PTFE matrix and Al particles debond, and cracks appear on the surface of the specimen under the action of tensile wave. When the strain rate reaches 3300 s −1 , the cracks generated by the tensile wave of the specimen expand and converge to form an open crack.  Figure 7 shows the specimens after dynamic compression at different strain rates at 20 • C. With the increase of strain rate, the radial size of the specimen increased gradually. When the strain rate is low, the size of the specimen does not change much after the dynamic compression test, while the end face of the specimen increases uniformly with the multiple loading of the stress wave, and the end face is relatively flat. When the strain rate is high, the energy carried by the stress wave is large. Before the specimen reaches the stress equilibrium state, it exceeds the deformation response limit of Al/PTFE as a viscoelastic material, resulting in local stress concentration. As a result, the center of the specimen is rapidly compressed, forming a pit. When the strain rate is 2800 s −1 , the PTFE matrix and Al particles debond, and cracks appear on the surface of the specimen under the action of tensile wave. When the strain rate reaches 3300 s −1 , the cracks generated by the tensile wave of the specimen expand and converge to form an open crack.  Table 2 shows the experimental parameters at different temperatures.

Effect of Temperature on Dynamic Compression Properties of Specimens
In order to determine the effect of temperature on the dynamic mechanical properties of the specimens, experiments No.26-No.31 were carried out. Table 2 shows the experimental parameters at different temperatures.  Figure 8 shows the stress-strain curves of the reinforced Al/PTFE specimens at the same strain rate and at different temperatures. It can be seen from the figure that with the decrease of temperature, the peak stress of specimens under the same strain rate gradually increases, and the material presents characteristics of temperature softening, while there is no obvious corresponding relationship between peak strain and temperature variation. The effects of the same temperature and different strain rates on the peak stress and peak strain are compared. The results are shown in Table 3. At the same temperature, the peak stress and peak strain increase with the increase of strain rate. At low temperature, the stress has declines significantly after the yield stage of the material, which is a result of the stress unloading process.

Effect of Temperature on Dynamic Compression Properties of Specimens
In order to determine the effect of temperature on the dynamic mechanical properties of the specimens, experiments No.26-No.31 were carried out. Table 2 shows the experimental parameters at different temperatures.  Figure 8 shows the stress-strain curves of the reinforced Al/PTFE specimens at the same strain rate and at different temperatures. It can be seen from the figure that with the decrease of temperature, the peak stress of specimens under the same strain rate gradually increases, and the material presents characteristics of temperature softening, while there is no obvious corresponding relationship between peak strain and temperature variation. The effects of the same temperature and different strain rates on the peak stress and peak strain are compared. The results are shown in Table 3. At the same temperature, the peak stress and peak strain increase with the increase of strain rate. At low temperature, the stress has declines significantly after the yield stage of the material, which is a result of the stress unloading process.

Microstructure Characterization and Dynamic Compression Process of Reinforced Al/PTFE
Scanning electron microscopy (SEM) was used to observe the microscopic morphology of the reinforced Al/PTFE specimens prepared in the same batch. The results are shown in Figure 9. The red box b in Figure 9 is the complete area of Figure 9, the red box c in Figure 9 is the complete area of Figure 9c, and the red box d in Figure 9 is the complete area of Figure 9. It can be seen from the figure that the Al particles were uniformly dispersed in the PTFE matrix without obvious agglomeration behavior. The reasons for some defects with apertures of 70-80 µm in the specimen are that PTFE has high crystallinity, and when the molding pressure is too large, the cold stretching generated by the powder particles may recover during the sintering process, resulting in the generation of micro-damages inside the material. The difference in thermal conductivity and thermal expansion coefficient between Al particles and the PTFE matrix will also lead to the existence of local large pores in specimens after sintering.

Microstructure Characterization and Dynamic Compression Process of Reinfo
Scanning electron microscopy (SEM) was used to observe the microsc ogy of the reinforced Al/PTFE specimens prepared in the same batch. T shown in Figure 9. The red box b in Figure 9 is the complete area of Figure c in Figure 9 is the complete area of Figure 9c, and the red box d in Figure 9 i area of Figure 9. It can be seen from the figure that the Al particles were persed in the PTFE matrix without obvious agglomeration behavior. The rea defects with apertures of 70-80 μm in the specimen are that PTFE has hig and when the molding pressure is too large, the cold stretching generated particles may recover during the sintering process, resulting in the genera damages inside the material. The difference in thermal conductivity and t sion coefficient between Al particles and the PTFE matrix will also lead to t local large pores in specimens after sintering. The dynamic deformation process from experiment No.12 was capture speed camera with a frame rate of 167,711 fps. According to the two-wav loading time of the first stress wave in No.12 was about 100 μs. In this p pression deformation images of the specimen in the first two stress wav cesses are selected, as shown in Figure 10. The previous frame's image of th of the specimen is defined as time t = 0 μs; (a)-(d) are the compression deform of the specimen loaded by the first stress wave pulse; (e)-(h) are the The dynamic deformation process from experiment No.12 was captured by the highspeed camera with a frame rate of 167,711 fps. According to the two-wave method, the loading time of the first stress wave in No.12 was about 100 µs. In this paper, the compression deformation images of the specimen in the first two stress wave loading processes are selected, as shown in Figure 10. The previous frame's image of the deformation of the specimen is defined as time t = 0 µs; (a)-(d) are the compression deformation process of the specimen loaded by the first stress wave pulse; (e)-(h) are the compression deformation process of the specimen loaded by the second stress wave. The results show that when the impact velocity is low, the overall deformation of the reinforced Al/PTFE specimen occurs under the action of stress wave. Since the friction force between the contact surface of the specimen and the bar cannot be completely eliminated, during the loading process, the radial stress existing on the end surface of the contact between the specimen and the bar hinders its radial deformation, which results in the specimen being drum-shaped during compression. deformation process of the specimen loaded by the second stress wave. The results show that when the impact velocity is low, the overall deformation of the reinforced Al/PTFE specimen occurs under the action of stress wave. Since the friction force between the contact surface of the specimen and the bar cannot be completely eliminated, during the loading process, the radial stress existing on the end surface of the contact between the specimen and the bar hinders its radial deformation, which results in the specimen being drumshaped during compression.
(e) t=420µs (f) t=480µs (g) t=540µs (h) t=600µs Direction of impact bar velocity Figure 10. Experiment No.12 dynamic compression process. Figure 11 shows the impact ignition process of the reinforced Al/PTFE specimen with the impact velocity of 35.2 m/s. Figure 11a-f shows the process of the specimen being gradually compressed to a pie shape. At t = 575 μs, although the diameter of the specimen has greatly exceeds that of the bar at this time, there are no cracks at the edge of the specimen. At this time, the diameter of the specimen is 23.7 mm, which is 1.69 times the initial diameter; the height of the specimen is 2.4 mm, which is 48% of the initial height. At t = 600 μs, the edge of the specimen cracks under the action of the tensile wave, but the overall structure is still relatively complete. At the same time, a shear band with a large strain gradient is formed in the contact area between the specimen and the edge of the compression bar. The material around the shear band undergoes an ignition reaction under the heat generated by plastic deformation and generates a bright spark [36].

(e) t=600µs (f) t=625µs (g) t=650µs (h) t=675µs
Direction of impact bar velocity 5mm Figure 11. The impact ignition process of reinforced Al/PTFE.  Figure 11 shows the impact ignition process of the reinforced Al/PTFE specimen with the impact velocity of 35.2 m/s. Figure 11a-f shows the process of the specimen being gradually compressed to a pie shape. At t = 575 µs, although the diameter of the specimen has greatly exceeds that of the bar at this time, there are no cracks at the edge of the specimen. At this time, the diameter of the specimen is 23.7 mm, which is 1.69 times the initial diameter; the height of the specimen is 2.4 mm, which is 48% of the initial height. At t = 600 µs, the edge of the specimen cracks under the action of the tensile wave, but the overall structure is still relatively complete. At the same time, a shear band with a large strain gradient is formed in the contact area between the specimen and the edge of the compression bar. The material around the shear band undergoes an ignition reaction under the heat generated by plastic deformation and generates a bright spark [36].
that when the impact velocity is low, the overall deformation of the reinforced Al/PTFE specimen occurs under the action of stress wave. Since the friction force between the contact surface of the specimen and the bar cannot be completely eliminated, during the loading process, the radial stress existing on the end surface of the contact between the specimen and the bar hinders its radial deformation, which results in the specimen being drumshaped during compression.
(e) t=420µs (f) t=480µs (g) t=540µs (h) t=600µs Direction of impact bar velocity Figure 10. Experiment No.12 dynamic compression process. Figure 11 shows the impact ignition process of the reinforced Al/PTFE specimen with the impact velocity of 35.2 m/s. Figure 11a-f shows the process of the specimen being gradually compressed to a pie shape. At t = 575 μs, although the diameter of the specimen has greatly exceeds that of the bar at this time, there are no cracks at the edge of the specimen. At this time, the diameter of the specimen is 23.7 mm, which is 1.69 times the initial diameter; the height of the specimen is 2.4 mm, which is 48% of the initial height. At t = 600 μs, the edge of the specimen cracks under the action of the tensile wave, but the overall structure is still relatively complete. At the same time, a shear band with a large strain gradient is formed in the contact area between the specimen and the edge of the compression bar. The material around the shear band undergoes an ignition reaction under the heat generated by plastic deformation and generates a bright spark [36].

Constitutive Equation of Reinforced Al/PTFE Active Materials at Room Temperature
From the stress-strain curves of the reinforced Al/PTFE material under quasi-static and dynamic compression, it can be seen that the material exhibits the characteristics of typical viscoelastic materials. In this paper, the mechanical behaviors of the reinforced Al/PTFE material are described based on the ZWT nonlinear viscoelastic constitutive model under different pressing pressures and different strain rates. The parameters of the constitutive equation of the material are fitted by the experimental results, and the accuracy of the model is verified.

ZWT Constitutive Model
The ZWT viscoelastic model at strain rates of 10 −4 -10 3 s −1 has been widely used in the study of polymer materials such as propellants, concretes, and organic glasses [37]. The rheological form of a ZWT nonlinear viscoelastic constitutive model consisting of a nonlinear spring and a low-frequency and high-frequency Maxwell viscoelastic element is shown in Figure 12.
From the stress-strain curves of the reinforced Al/PTFE material under quasi-static and dynamic compression, it can be seen that the material exhibits the characteristics of typical viscoelastic materials. In this paper, the mechanical behaviors of the reinforced Al/PTFE material are described based on the ZWT nonlinear viscoelastic constitutive model under different pressing pressures and different strain rates. The parameters of the constitutive equation of the material are fitted by the experimental results, and the accuracy of the model is verified.

ZWT Constitutive Model
The ZWT viscoelastic model at strain rates of 10 −4 -10 3 s −1 has been widely used in the study of polymer materials such as propellants, concretes, and organic glasses [37]. The rheological form of a ZWT nonlinear viscoelastic constitutive model consisting of a nonlinear spring and a low-frequency and high-frequency Maxwell viscoelastic element is shown in Figure 12. The basic form of the ZWT constitutive model is where ( ) For quasi-static and low strain-rate loading conditions, the influence of the high strain-rate Maxwell element is ignored [38]. The quasi-static loading is a constant strain rate loading, / t ε ε′ = , and the equation is simplified to where f e (ε) describes the nonlinear elastic response of the equilibrium state independent of the strain rate, E 0 is the initial elastic modulus, and α and β are the corresponding elastic constants. The second and third integral terms in Formula (6) describe the low strain-rate and high strain-rate viscoelastic responses of two Maxwell elements, respectively. E 1 and θ 1 are the elastic modulus and the stress relaxation time of the low-frequency Maxwell element, E 2 and θ 2 are the elastic modulus and the stress relaxation time of the highfrequency Maxwell element. Since the stress-strain curve of the quasi-static compression experiment is linear before the yield stage, the equilibrium stress f e (ε) which is independent of the strain rate can be considered as E 0 ε only. For quasi-static and low strain-rate loading conditions, the influence of the high strainrate Maxwell element is ignored [38]. The quasi-static loading is a constant strain rate loading, t = ε/ε , and the equation is simplified to Under high strain-rate loading conditions, the low-frequency Maxwell element and the two parallel springs describing the equilibrium stress element are reduced to a single spring element. SHPB experiment is constant strain-rate loading, t = ε/ε , and the equation is simplified to

Determination of Constitutive Model Parameters
Due to the quasi-static loading, the static compressive stress-strain curve of reinforced Al/PTFE at the same strain rate was little affected by the pressing pressure factor. The stress-strain curve obtained from the quasi-static compression experiment with the pressing pressure of 400 MPa at room temperature was selected for fitting, and E 0 , E 1 and θ 1 were obtained. The fitting results are shown in Figure 13. The correlation coefficient R 2 of the curve fitting is 0.98. The fitting results are E 0 = 123 MPa, E 1 = 224 MPa, and θ 1 = 83.7 s.   The stress-strain curves of No.14 and No.22 with different strain rates are fitted so as to obtain the elastic constant E 2 and relaxation time θ 2 , which are suitable for strain rates ranging from 1400 s −1 to 2600 s −1 when the pressing pressure is 600 MPa. The fitting results are shown in Figure 15. The correlation coefficients R 2 of curve fitting are 0.99 and 0.94, respectively. The fitting results are E 2 = 4.9 GPa, θ 2 = 4.06 µs. The parameters of the ZWT constitutive model obtained by different pressing pressures have been determined, and the parameters are listed in Table 4. The parameters of the ZWT constitutive model obtained by different pressing pressures have been determined, and the parameters are listed in Table 4.  19, which were not involved in the model fitting, and whose strain rate is 2000 s −1 , were used to test the ZWT constitutive model considering the pressing pressure in the range of 1400-2600 s −1 . The results are shown in Figure 17. The correlation coefficients R 2 of curve fitting are 0.98, 0.94 and 0.99, respectively. In the corresponding strain rate range, the experimental curves are in good agreement with the predicted curves. The model can reliably describe the mechanical behavior of the reinforced Al/PTFE under the influence of pressing pressure factors in the strain rate range of 1400-2600 s −1 .   19, which were not involved in the model fitting, and whose strain rate is 2000 s −1 , were used to test the ZWT constitutive model considering the pressing pressure in the range of 1400-2600 s −1 . The results are shown in Figure 17. The correlation coefficients 2 R of curve fitting are 0.98, 0.94 and 0.99, respectively. In the corresponding strain rate range, the experimental curves are in good agreement with the predicted curves. The model can reliably describe the mechanical behavior of the reinforced Al/PTFE under the influence of pressing pressure factors in the strain rate range of 1400-2600 s −1 .

Constitutive Equation of Reinforced Al/PTFE Active Materials at Low Temperature
The traditional ZWT constitutive model does not consider the influence of temperature on the material, which is slightly insufficient. Wang et al. [39] modified the ZWT constitutive model by adding temperature correlation terms, and described the mechani-

Constitutive Equation of Reinforced Al/PTFE Active Materials at Low Temperature
The traditional ZWT constitutive model does not consider the influence of temperature on the material, which is slightly insufficient. Wang et al. [39] modified the ZWT constitutive model by adding temperature correlation terms, and described the mechanical properties of rubber at different temperatures and strain rates. Zhang et al. [40] improved the ZWT constitutive model based on temperature-dependent parameters and established a thermoviscoelastic constitutive model to describe the mechanical properties of polyurethane films at different temperatures and strain rates. In order to describe the effect of the temperature on the mechanical properties of reinforced Al/PTFE materials, the constitutive model parameters were regarded as temperature-dependent functions.

Establishment of the Constitutive Model
In order to describe the effect of temperature on the mechanical properties of reinforced Al/PTFE materials, the temperature parameter was introduced into the equation, and a simple thermal viscoelastic constitutive model was obtained.

Determination of Constitutive Model Parameters
In order to obtain the thermal viscoelastic constitutive parameters of the reinforced Al/PTFE materials, the stress-strain curves obtained by quasi-static compression experiments at three temperatures (−20 • C, −30 • C, and −40 • C) were fitted to obtain the elastic modulus E 0 and E 1 of the linear springs at three temperatures. The fitting results are shown in Figure 18, and the correlation coefficients R 2 of curve fitting are 0.97, 0.97, and 0.99, respectively. In order to describe the effect of temperature on the mechanical properties of reinforced Al/PTFE materials, the temperature parameter was introduced into the equation, and a simple thermal viscoelastic constitutive model was obtained.

Determination of Constitutive Model Parameters
In order to obtain the thermal viscoelastic constitutive parameters of the reinforced Al/PTFE materials, the stress-strain curves obtained by quasi-static compression experiments at three temperatures (−20 °C, −30 °C, and −40 °C) were fitted to obtain the elastic modulus 0 E and 1 E of the linear springs at three temperatures. The fitting results are shown in Figure 18, and the correlation coefficients   Table 5. The elastic moduli E 0 and E 1 of the linear spring obtained from the quasi-static compression experiment were substituted into Equation (10), and the elastic constant E 2 and relaxation time θ 2 in the parameters of the ZWT constitutive model were obtained by fitting the experimental curves in each high strain-rate range at different temperatures. Figure 19, Figures 20 and 21 shows the fitting results. The parameters of the constitutive model are listed in Table 5.    The parameters of the ZWT constitutive model in Table 5 are fitted concerning tem-     The parameters of the ZWT constitutive model in Table 5 are fitted concerning temperature factors in order to obtain the relationship between each parameter value and temperature, and the fitting results are shown in Table 6.     The parameters of the ZWT constitutive model in Table 5 are fitted concerning temperature factors in order to obtain the relationship between each parameter value and  The parameters of the ZWT constitutive model in Table 5 are fitted concerning temperature factors in order to obtain the relationship between each parameter value and temperature, and the fitting results are shown in Table 6. Under the conditions of different loading strain rates, the yield strength of the reinforced Al/PTFE material decreases with the increase in temperature, which has a thermal softening effect. Referring to the basic form of the Johnson-Cook constitutive model [41], based on the above analysis, a simple thermal viscoelastic constitutive model was obtained by adding the thermal softening terms.
where m is the thermal softening index, T is the current temperature, T m is the melting point of the material (set as the melting point of PTFE matrix 402 • C), and T r is the reference temperature (set as the lowest experimental temperature −40 • C).

Determination of Constitutive Model Parameters
The ZWT constitutive model parameters at −40 • C were substituted into Formula (11), and the thermal softening index m at different temperatures was obtained by fitting the dynamic stress-strain curves of the reinforced Al/PTFE at different temperatures. The fitting results are shown in Figures   Under the conditions of different loading strain rates, the yield strength of the reinforced Al/PTFE material decreases with the increase in temperature, which has a thermal softening effect. Referring to the basic form of the Johnson-Cook constitutive model [41], based on the above analysis, a simple thermal viscoelastic constitutive model was obtained by adding the thermal softening terms.
where m is the thermal softening index, T is the current temperature, m T is the melting point of the material (set as the melting point of PTFE matrix 402 °C), and r T is the reference temperature (set as the lowest experimental temperature −40 °C).

Determination of Constitutive Model Parameters
The ZWT constitutive model parameters at −40 °C were substituted into Formula (11), and the thermal softening index m at different temperatures was obtained by fitting the dynamic stress-strain curves of the reinforced Al/PTFE at different temperatures. The fitting results are shown in Figures

Conclusions
Based on the traditional formula Al/PTFE (26.5%/73.5%), the reinforced Al/PTFE active material was prepared by cold pressing and sintering combining with rapid cooling. Quasi-static and dynamic compression experiments were conducted on the reinforced Al/PTFE specimens prepared under different compression pressures and at different temperatures and strain rates. Based on the compression experimental data, the parameters of the Zhu-Wang-Tang (ZWT) constitutive model of the reinforced Al/PTFE active material were fitted. The ZWT constitutive model parameters of the reinforced Al/PTFE active material under different compression pressures at room temperature and the ZWT constitutive model parameters of the reinforced Al/PTFE active material at low temperature were obtained. The accuracy of constitutive model parameters (elastic modulus, stress relaxation time, and thermal softening index) was verified. The ZWT constitutive model established in this paper considers both the temperature and the strain rate, making up for the shortcomings of the classical ZWT model (lack of a temperature term). The strength of this kind of active material has been greatly improved compared with the traditional process, which can be widely used in warhead shells, shaped charge liners, and other damage elements in order to meet the high efficiency damage demand for high-value targets. In the future, the application scope of temperature and strain rate will be broadened in order to improve the universality of the constitutive model. This paper provides theoretical and data support for the establishment of the ZWT constitutive model of active materials applicable to a wider range of temperature and strain rate.

Conclusions
Based on the traditional formula Al/PTFE (26.5%/73.5%), the reinforced Al/PTFE active material was prepared by cold pressing and sintering combining with rapid cooling. Quasi-static and dynamic compression experiments were conducted on the reinforced Al/PTFE specimens prepared under different compression pressures and at different temperatures and strain rates. Based on the compression experimental data, the parameters of the Zhu-Wang-Tang (ZWT) constitutive model of the reinforced Al/PTFE active material were fitted. The ZWT constitutive model parameters of the reinforced Al/PTFE active material under different compression pressures at room temperature and the ZWT constitutive model parameters of the reinforced Al/PTFE active material at low temperature were obtained. The accuracy of constitutive model parameters (elastic modulus, stress relaxation time, and thermal softening index) was verified. The ZWT constitutive model established in this paper considers both the temperature and the strain rate, making up for the shortcomings of the classical ZWT model (lack of a temperature term). The strength of this kind of active material has been greatly improved compared with the traditional process, which can be widely used in warhead shells, shaped charge liners, and other damage elements in order to meet the high efficiency damage demand for high-value targets. In the future, the application scope of temperature and strain rate will be broadened in order to improve the universality of the constitutive model. This paper provides theoretical and data support for the establishment of the ZWT constitutive model of active materials applicable to a wider range of temperature and strain rate.

Conclusions
Based on the traditional formula Al/PTFE (26.5%/73.5%), the reinforced Al/PTFE active material was prepared by cold pressing and sintering combining with rapid cooling. Quasi-static and dynamic compression experiments were conducted on the reinforced Al/PTFE specimens prepared under different compression pressures and at different temperatures and strain rates. Based on the compression experimental data, the parameters of the Zhu-Wang-Tang (ZWT) constitutive model of the reinforced Al/PTFE active material were fitted. The ZWT constitutive model parameters of the reinforced Al/PTFE active material under different compression pressures at room temperature and the ZWT constitutive model parameters of the reinforced Al/PTFE active material at low temperature were obtained. The accuracy of constitutive model parameters (elastic modulus, stress relaxation time, and thermal softening index) was verified. The ZWT constitutive model established in this paper considers both the temperature and the strain rate, making up for the shortcomings of the classical ZWT model (lack of a temperature term). The strength of this kind of active material has been greatly improved compared with the traditional process, which can be widely used in warhead shells, shaped charge liners, and other damage elements in order to meet the high efficiency damage demand for high-value targets. In the future, the application scope of temperature and strain rate will be broadened in order to improve the universality of the constitutive model. This paper provides theoretical and data support for the establishment of the ZWT constitutive model of active materials applicable to a wider range of temperature and strain rate.