Dynamic compression intrinsic relationship of W-Ni-Fe-Mn alloys based on J-C modeling

Dynamic compression tests were performed on the sample alloys by using a detached Hopkinson compression bar with a diameter of 14.5 mm caliber for the kinetic behavior of W-Ni-Fe alloy and W-Ni-Fe-Mn alloy materials in the erosion process. The true stress-true strain curves of the two sample alloys were fitted based on the JC eigenstructure model, and the eigenstructure relationships of the two alloys were finally obtained. The obtained constitutive equations were then used to dynamically compress the specimens by using LS-DYNA dynamics software, and the simulated true stress-strain curves of the two alloys at various strain rates agreed well with the experimentally obtained curves, allowing them to be used for simulation calculations under impact loading. When compared toW-Ni-Fe alloys, the dynamic mechanical characteristics of alloys with Mn addition are greatly improved.


Introduction
The use of tungsten alloy in the fabrication of armor-piercing bullet cores is prevalent due to its exceptional attributes, including elevated strength, substantial density, and robust corrosion resistance.Armor-piercing ammunition is a type of munition that utilizes its inherent kinetic energy to effectively penetrate armored targets.This ammunition possesses several advantageous characteristics, including high velocity, formidable penetration capabilities, and exceptional accuracy.As a result, it finds extensive application in combat scenarios involving armored targets such as tanks, armored vehicles, and low-altitude helicopter gunships.The efficacy of this ammunition is quite potent, as it possesses the capability to effectively dismantle the adversary's equipment, assuming a pivotal part in securing triumph in the conflict.Currently, the alloy warhead often employed in armor-piercing ammunition is typically composed of tungsten-nickel-iron (W-Ni-Fe) alloy [1] .
However, this warhead is susceptible to developing a mushroom-shaped head during the armorpiercing process, mostly due to its insensitivity to adiabatic shear.Consequently, the capacity of the warhead to penetrate armor is greatly diminished.The newly developed alloy, composed of tungsten, nickel, iron, and manganese (W-Ni-Fe-Mn), exhibits enhanced adiabatic shear capabilities and possesses a self-sharpening characteristic during the armor-piercing process while maintaining its penetration depth.Consequently, this alloy holds promise as a potentially superior alternative to depleted uranium alloys.
The Johnson-Cook primary model, initially suggested by Johnson and Cook in 1983, is a semiempirical model that draws upon experimental phenomena.It has demonstrated efficacy in various domains, including armor-piercing mechanics.The precise characterization of the dynamic mechanical characteristics of materials can be achieved through a complete analysis of the impacts of strain hardening, strain rate hardening, and material softening on these properties.σ =(A+Bɛ n )(1+έ*) c (1-T* m ) [7] (1) Dynamic compression of hot-pressed and sintered W-Ni-Fe alloys and W-Ni-Fe-Mn alloys is performed in this work by utilizing a split Hopkinson pressure bar to collect stress-strain curves at various strain rates and to fit the MJC constitutive equations of the two alloys [3] .Finally, the dependability of the constitutive equations is tested by entering the generated constitutive relations into the finite element program LS-DYNA and running the calculation.

Dynamic compression analysis
The use of a separate Hopkinson press bar (SHPB) is currently a popular method for investigating the dynamic mechanical properties of materials.The schematic diagram of the separated Hopkinson press rod device used in this experiment is shown in Figure 1.The specifications of the separated Hopkinson press rod device used in this experiment are: both lengths are 200 mm, 14.5 mm incident rod, transmission rod, and reflection rod; the test system is composed of the LK2107A ultra-dynamic strain gauges and oscilloscopes.The test principle is driven by a high-pressure gas bullet impact on the incident rod, and the incident pulse is transmitted along the rod in the incident rod.When the pulse is transmitted to the specimen, high-speed deformation occurs [5] , and a portion of the pulse energy is absorbed by the deformation of the specimen.Another portion propagates to the projection rod to form a projection pulse, and the remaining portion is reflected in the incident rod.The link between strain and strain rate, one-dimensional stress assumption, and homogeneity assumption can be obtained.

Figure 1 Schematic diagram of split hopkinson pressure bar (SHPB).
The composition of the two alloys is shown in Table 1.W-Ni-Fe and W-Ni-Fe-Mn alloy specimens were generated for this experiment by utilizing the mechanical alloying method and two-step sintering via the ball milling wet mixing-solid phase, sintering-liquid phase, and sintering-post-sintering heat treatment (vacuum annealing). [1]The sintered alloys were sliced into 3 mm*3 mm samples by using wire cutting, and the samples were then examined for dynamic mechanical characteristics Table 1 Alloy composition.

JC model ontological relationships
According to the above, it can be seen that we use the modified MJC intrinsic mode pattern as follows: σ = (A+Bɛn)(1+ɛ*)c(1-T*m), where A, B, n, C, and m are the material parameters, which need to be corrected by the experimental fitting, and έ*=έ/έ0, where έ denotes the current strain rate, and έ0 denotes the reference strain rate, which is taken as the value of 1000/s.Since the experiments were performed at room temperature, T-Tm = 0, and the thermal softening term balance is 1.The effect due to temperature is neglected and m does not need to be fitted.
In the MJC constitutive relationship, the parameter A represents the yield strength of the material at rest, and σ0 represents the yield strength of the material at each strain rate.When the plastic strain of the material is 0, Equation ( 2) can be written in the following form: σ=A(1+έ* ) C (2) While it is difficult to obtain an accurate yield strength by dynamic compression of the sample, we use a type of bilinear crossover method with elastic and plastic segments in this study to obtain an approximate yield strength at each strain rate.Figure 2 depicts the yield strengths of W-Ni-Fe alloy and W-Ni-Fe-Mn alloy at various strain rates, which are then fitted to obtain the hardening [6] .
The term of strain rate in the constitutive equations is as follows: W-Ni-Fe alloy: σ0= 1358(1+έ*) 0.016 W-Ni-Fe-Mn alloy: σ0=1367(1+έ*) 0.016 (4) After obtaining the two parameters A and C, the true stress-strain curves at each strain rate were written in the form of Equation ( 5

The MJC intrinsic model's metric validation
LS-DYNA is the dynamics simulation program.In this paper, the above-mentioned MJC constitutive relationship is entered into the finite element program LS-DYNA for simulation computation, and the dynamic impact compression simulation model of the prototype size is developed by the SHPB experiment.The settings were as follows: 0.9 mm radial and axial meshes for incident and projectile rods, 5 mm axial mesh size, 0.15 mm specimen mesh size, and the elastic model MAT_ELASTIC for incident and projectile rods and bullets.Because the specimen data processing is based on the sample's homogeneous deformation, the specimen's center unit is used to produce its axial stress-strain curve for comparison with the experimental curve.For the simulation computations, the bullet velocity is changed so that it matches each material at each strain rate.Because the experimental data processing is based on the homogeneous deformation of the specimen, the axial stress-strain at the site of the center cell is used for simulation data analysis and compared to the experimental curve.
Because the specimen data processing is based on the sample's homogeneous deformation, the specimen's center unit is used to output its axial stress-strain versus the experimental curve (Figure 4).
Figures 3 and 4 compare the simulation curves and experimental curves of W-Ni-Fe alloy and W-Ni-Fe-Mn alloy at different strain rates, and the figures show that the simulation results agree well with the experimental results, indicating that the parameters of the MJC intrinsic model under dynamic compression are highly reliable [2] .

Effect of Mn on the microstructure of alloys
Figure 5 presents the metallographic specimens, allowing for a comparison between (a) and (b).The introduction of the Mn element results in a gradual transformation of the W grain into a virtually spherical shape.Furthermore, a comparison between (c) and (d) reveals a reduction in the size of the W grain from a range of 20~45 μm in the absence of Mn to 12~15 μm.The liquid-phase sintering mechanism of W-Ni-Fe tungsten alloy involves the creation of a Ni-Fe liquid phase and the dissolution and subsequent re-precipitation of tungsten [4] .Additionally, when tungsten alloys with manganese (Mn) are sintered, the Ni-Fe liquid phase contains manganese oxide (MnO), which enhances the nucleation rate of tungsten particles.Simultaneously, the presence of MnO in the liquid phase impedes the diffusion of tungsten during the dissolution and re-precipitation process, hence diminishing the rate at which tungsten particles grow.Consequently, the incorporation of manganese serves to enhance the size reduction of tungsten particles.

Effect of Mn on mechanical properties of alloy gold
Figure 6 illustrates the comparative hardness relationship curves of the two alloys.It is observed that the inclusion of the Mn element increases the relative density of the alloy.Furthermore, the average microhardness rises from 371 HRC to 412 HRC.
According to Walsh, it is suggested that the presence of spherical micropores has a limited impact on the mechanical characteristics of alloys with greater specific gravity.Conversely, the presence of coarse macropores or dissolved gases is shown to result in a reduction in alloying qualities.This observation is supported by the data presented in Figure 5.The voids observed in graph (a) are comparatively smaller than those in graph (b), indicating an increase in the Vickers hardness of the alloys.This can be attributed to the reactive nature of the chemical properties of manganese (Mn), which exhibits a high affinity for oxygen and sulfur.The generation of manganese oxide during the alloy formation process has the effect of purifying the interfaces, contributing to the observed increase in hardness.
The manganese oxide produced exhibits the ability to enhance the purification of the interface, leading to improved wettability and bonding characteristics of tungsten (W) and nickel-iron (Ni-Fe), consequently resulting in an increase in the hardness of the alloy.

Conclusions
This paper examines the dynamic mechanical properties of W-Ni-Fe alloys and W-Ni-Fe-Mn alloys through the utilization of the separated Hopkinson compression bar.Furthermore, the MJC intrinsic model is established for W-Ni-Fe alloys based on the original JC model.Subsequently, the parameters of the fitted intrinsic model are incorporated into the simulation model to conduct numerical simulations of impact compression.The ensuing conclusions have been derived from this investigation: (1) The intrinsic parameters of the two alloys, W-Ni-Fe alloy and W-Ni-Fe-Mn alloy, were determined by fitting the experimental data.For the W-Ni-Fe alloy, the values obtained were A=1358 MPa, B=338 MPa, n=0.25, C=0.016, and m=1.Similarly, for the W-Ni-Fe-Mn alloy, the values obtained were A=1367 MPa, B=340 MPa, n=0.30,C=0.016, and m=1.
(2) The stress-strain curves derived by modeling impact compression using the fitted constitutive equations exhibit a high level of concordance with the experimental findings.The findings of this study demonstrate a high level of dependability in the fitted dynamic compression constitutive equations for W-Ni-Fe alloys and W-Ni-Fe-Mn alloys, as determined by using the MJC model.
(3) Based on the obtained real stress-strain curves, a comparative analysis of the fitted curves for W-Ni-Fe and W-Ni-Fe-Mn alloys reveals that the incorporation of Mn components leads to enhanced dynamic mechanical capabilities in the alloys.
(4) The introduction of the Mn element has resulted in a reduction of the W grain size from 20-45 μm to 12-15 μm, thereby serving as a means of grain refinement.
(5) Following the introduction of the Mn element, the micro-mean hardness of the alloy exhibited a notable rise, rising from 371 HRC to 412 HRC.

Figure 2
Figure 2 Yield strength of alloys at different strain rates.(a) W-Ni-Fe alloy; (b) W-Ni-Fe-Mn alloy.