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Parameter Id of Metal Hi-pressure State Equation

Publié en ligne: 31 May 2022
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 18 Jan 2022
Accepté: 27 Mar 2022
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Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
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2 fois par an
Langues
Anglais
Abstract

In this study, parameters of the Grüneisen equation of state for GH4169 alloy were calculated based on multi-scale impact technology and first-principles calculation methods. The calculated parameters are consistent with the results of Liu et al., which primarily verifies the accuracy of the method. The AUTODYN software was used for numerical simulation of dynamic plate impact experiments. The parameters of the Grüneisen equation of GH4169 alloy were used as input to verify its accuracy. Comparing and analysing the speed of the free surface particle and the actual experimental measurement point at the same position, it is concluded that the simulated value is consistent with the experimental value. The morphology of the flying piece and the target have the same characteristics, which proves that Grüneisen equation of state parameters obtained by proposed parameter identification method are practical and reliable.

Keywords

Introduction

Metal materials are widely used in weaponry, aerospace and transportation fields. The equation of state at high temperature and pressure is the basis for studying properties of metal materials. The equation of state is an equation that expresses the functional relationship between the independent parameters of the equilibrium macroscopic properties of a uniform system and other state parameters [1, 2]. It generally refers to the functional relationship between system pressure P, temperature T (or internal energy E) and volume (or specific volume) V. Metal materials will fail or suffer serious damage under high temperature and high pressure environment, which has a certain relationship with their performance. The performance research of metal materials under high temperature and pressure environment mainly depends on the equation of state to judge the law of performance change. The Grüneisen equation of state can be combined with finite element software to study the change of material properties under extreme conditions, so its application is very wide [3]. Therefore, it is of great significance to explore the identification method of parameters of Grüneisen equation of state.

GH4169 nickel-based superalloy was selected as the research object with two reasons. First, GH4169 alloy is mainly used in the hot end parts of industrial gas turbines and aero-engines, and its working environment is usually high temperature, high pressure and impact load. Its performance is related to the life cycle and safety of the product. Therefore, obtaining the parameters of the Grüneisen equation of state has important meaning for the application of the material in the aerospace field. Second, as an alloy material containing nearly ten kinds of elements, GH4169 nickel-based superalloy is extremely difficult to develop its potential function, and there are many precipitation phases in the alloy. Therefore, a Grüneisen state equation parameter identification method that can break through the restriction of the potential function is needed.

Literature Review

With the increasing demand for complex alloys with excellent properties in the fields of aerospace and weaponry, the circumstances or environment where we use metal materials has become more demanding. For example, an aero engine is always in a state of high temperature and high pressure during the working state. Therefore, the impact of alloy properties under extreme service environments must be considered when selecting materials.

Both isothermal and adiabatic equations of state have made great progress. There are two types of isothermal state equations: one is to draw a cold curve (0 K), and the other is at 300 K. Bridgman used a static compression device when studying the compression properties of materials under high pressure. Studies have shown that the heat generated inside the material can achieve complete exchange with the outside air due to the very slow compression process. Therefore, this compression process could be approximated as isothermal compression [4]. Many scholars have conducted high-pressure isothermal compression studies on materials [5,6,7,8]. Syassen and Holzapfel [9] used 12 GPa x-ray diffraction to study the isothermal compression process and curve changes of Ag and Al. In their research, the Birch equation was used as an isothermal state equation to compare with experimental data. Vočadlo et al. [10] and Shanker et al. [11] have explored issues of isothermal body modulus and first-order pressure derivative by comparing various isothermal state equations. Hrubiak et al. [12] explored the isotherm of Hf at room temperature. They obtained the P(E, V)-type equation of state for α-phase Hf at a pressure of 67 GPa by using two-parameter Birch-Murnaghan equation. Akahama et al. [13] conducted an x-ray diffraction experiment where Birch-Murnaghan and Vinet equations of state were used to describe the 300K isotherms of Bi, Pt and Au.

Adiabatic equations of state are usually based on Hugoniot data. Hugoniot data comes from a flying debris impact experiment powered by a light airgun or explosive. Al’e Tshuler et al. [14] studied the equations of state of Al, Cu and Pb at high pressure environment. Bringa et al. [15] and Agarwal et al. [16] explored the linear relationship between the downward shock velocity and particle velocity of different crystals by analysing the non-equilibrium molecular dynamics (NEMD) of the adiabatic shock line of Mg and Cu single crystals. Mackenchery and Dongare [17] analysed the impact of four different atomic potentials on Ti single crystal at collision velocities and Hugoniot curves of approximately 0.5–2.0 km/s based on molecular dynamics (MD).

As a solid model under high pressure, Mie-Gruneisen equation of state has been widely used in various fields. In many simulation software, the parameters of Mie-Gruneisen equation of state could be directly inputted to define the material. Walsh and Christian [18] Walsh et al. [19] studied the equation of state of metals through a blasting system, a measuring volume dependence for Gruneisen coefficient γ. Mc Queen and Marsh [20] explored the equations of state of 19 metal elements in a blasting system which include a plane wave producer with a pressure of approximately 100–200 GPa. The Hugoniot relation (P, V, E) is generalised to a more general equation of state (E P, V, T) based on Mie-Gruneisen theory. Nellis et al. indicate the pressure range of copper and Ta Hugoniot curves at 80–440 gpa and discuss the influence of temperature and γ [21]. Jin et al. [22] proposed a thermodynamic formula, based on adiabatic shock line data, to characterise the 0K isotherms of metals, and predicted the 0K isotherms and the Gruneisen equation of state of seven metals by using Born-Meyer potential. Yang et al. [23] studied Hugoniot curve, isotherm curve, isentropic curve and internal energy using molecular dynamics method. Molecular dynamics simulation results showed Mie-Gruneisen equation of state for Al and Pb single crystals. In his study, Yang calculated and discussed that Hugoniot curve and isentropic-line increase along temperature and entropy along Hugoniot curve, and Gruneisen coefficient γ is consistent with existing literature. Zhang et al. proposed a new temperature-dependent energy thermodynamic model to study the temperature dependence of materials Grüneisen. Molecular dynamics simulation of Ni, Cu and Au by molecular dynamics verified the newly established thermodynamic model [25].

For isotherms, Hugoniot curve and Mie-Grüneisen equation of state have been studied extensively. The types of materials studied mainly focus on single element and binary alloys, while the research on multi-element alloys is relatively rare. As we all know, the physical properties of composite alloys are very different from the physical properties of their constituent elements. They are usually used in more complex environments. Therefore, it is of great significance to explore dynamic mechanical properties of complex alloys under extreme conditions. However, due to the lack of corresponding potential functions, parameters of Mie-Gruneisen equation of state for multi-element alloys are difficult to obtain through molecular dynamics simulations. This research proposes a parameter identification method for calculating parameters of Mie-Gruneisen equation of state for multi-element alloys according to Ji et al. [26] and Millett et al. [27].

Method for identifying parameters of high-pressure equation of state for metallic materials

Due to the addition of a variety of elements to improve various aspects of performance, multi-element alloys have a wider range of applications than general metal materials. Due to the excellent performance of multi-element alloys, the service environment is generally extreme. The study of the equation of state mainly relies on theoretical analysis, experiment and molecular dynamics simulation. However, the experiment will cost too much. Equilibrium dynamics is not suitable for dynamic analysis, and it takes a lot of time to calculate large-scale models for non-equilibrium dynamics. Reed et al. proposed Multi-scale Shock Technique (MSST) that is not only overcome these problems, but also reduce the model size while maintaining the simulation accuracy, thereby effectively reducing the calculation time [33].

MSST refers to the molecular dynamics simulation of a small part of the shock wave at a given transient time. Adjusting the applied stress and energy can be used to simulate the shock wave effect of this small molecule dynamics system. These stresses and energies satisfy the continuity theory describing the shock wave structure [30, 31]. This method ensures that the thermodynamic path is correctly captured, and the (P, T) thermodynamic state simulated by the shock wave is used to obtain a result consistent with the stable macroscopic shock wave. Compared with non-equilibrium molecular dynamics, MSST significantly reduces the number of particles and ensures that the simulation correctly converges to the thermodynamic state [32]. MSST makes it possible to simulate with fewer atomic shock waves, which is of great significance. According to Ji et al. [26] and Millett et al. [27], we propose a method for parameter identification of the Grüneisen equation of state for multi-element alloys.

GH4169 alloy

The chemical composition of GH4169 alloy is a complex one, and the mass fraction of many trace metal elements is <1%. Therefore, we have simplified the composition of GH4169 to a certain extent. First, retain the metal elements with a mass fraction >1%. The elements that meet this condition are Ni, Cr, Fe, Nb, Mo and Ti. Secondly, because GH4169 nickel-based alloy has γ′ phase, it is of great significance to enhance the high temperature performance of GH4169. Therefore, all metal elements contained in γ′ phase should also be retained. The elements satisfying this condition are Ni, Nb, Ti and Al. The simplified mass fractions of the main elements of GH4169 alloy are shown in Table 1.

Mass fraction and atomic fraction of each element of GH4169 alloy

Element Mass fraction (%) Atomic fraction (%)

Ni 53.40 52.40
Cr 18 20.80
Fe 18.20 19.50
Mo 3.50 1.40
Nb 4.80 2.60
Ti 1.50 1.90
Al 0.60 1.40
Hugoniot curve parameters of GH4169 alloy based on molecular dynamics

Molecular dynamics (MD) is a numerical simulation method for studying scientific problems at the atomic scale [29]. In molecular dynamics simulation, the motion of all particles satisfies Newton's second law. By solving the Newtonian differential equation of motion, the velocity and position of any atom in the system can be obtained.

Fi(t)=miai(t)=mi2riti2 {F_i}(t) = {m_i}{a_i}(t) = {m_i}{{{\partial ^2}{r_i}} \over {\partial t_i^2}}

Fi(t) is the force exerted by other atoms on atom i at t; ai(t) represents the acceleration of atom i at t; ri is the position vector of the atom; mi is the mass of the atom. When the total energy of the system is U, the force Fi(t) exerted by other atoms on the atom i at time t can be expressed as the position gradient of the total energy U.

Fi(t)=U {F_i}(t) = - \nabla U

Any particle in the system satisfies at time t: mi2riti2=U {m_i}{{{\partial ^2}{r_i}} \over {\partial t_i^2}} = - \nabla U

After determining the potential function used by the research body, the trajectory of each atom can be obtained by analysing the above Newton equation, and the coordinates, acceleration, particle velocity and other information at any time can be obtained. Then, based on the obtained information, the macroscopic physical quantities of the system (such as pressure, energy, temperature, radial distribution function) can be calculated through statistics and averaging, and the relevant properties of the material can be explained.

The size of all models is 20a×20a×20a, where a is the lattice constant of a single unit cell. In each direction, periodic boundary conditions are used. Before loading the shock, relax the model in the NPT system by 40 ps at 300 K, making it in a state where the residual stress can be ignored. After relaxation, a shock wave is applied in the X-axis direction of the model through MSST. The wave speed of Ni is 11–15 km/s, the wave speed of Ni3Nb is 10–18 km/s, the wave speed of Ni3Ti is 6–10 km/s, the wave speed of Ni3Al is in the range is 6–10 km/s, the wave speed range of Cr is 11–15 km/s, the wave speed range of Fe is 6–10 km/s and the wave speed range of Mo is 6–10 km/s. The particle's velocity, pressure, temperature and other physical quantities will stabilise when the simulation lasts for a long enough time to extract the required data from the data in the steady state.

C0 and λ of GH4169 alloy

In GH4169 alloy, the γ′ phase is Ni3Nb, which is the main strengthening phase. The γ′ phase includes Ni3Al and Ni3Ti, which are auxiliary strengthening phases; Ni is the γ phase, which is the matrix. Among the phases involved in this paper, there is no δ phase. According to the relevant heat treatment system, the heat treatment method for GH4169 without δ phase is 1010–1065°C±10°C for 1 hour; oil cooling at 720°C±5°C for 8 hours; at 50°C/h furnace cooling to 620°C±5°C for 8 hours, air cooling. After calculating based on the atomic ratio, the ideal mass fraction of each phase can be obtained. The C0 and λ of each phase were obtained by MSST (Table 2).

Mass fraction and Hugoniot parameters of each phase in GH4169 alloy

Phase Mass fraction (%) C0 λ

Ni 35 4.4133 1.1419
Cr 20 4.9712 1.1196
γ-Fe 19 3.8327 2.2667
Mo 5.00 5.9806 0.9444
γ″-Ni3Nb 12.60 5.5252 2.6675
γ″-Ni3Ti 4.80 5.6461 1.0397
γ″-Ni3Al 3.60 5.3751 1.068

Table 3 shows the C0 and λ of the GH4169 nickel-based superalloy obtained by fitting. By comparing with the research results of Liu et al., the C0 and λ calculated by the theoretical calculation method proposed in this study are compared with those obtained by the first grade lighter. The C0 and λ obtained from the flat plate impact experiment of the gas gun are very close to the theoretical values. It proves that the calculation theory proposed in this research is correct.

Hugoniot parameters and relative error of GH4169 alloy

C0 λ

Hugoniot parameters in this research 4.532 1.62
Liu et al. [34] 4.485 1.45
Relative error 0.70% 15.60%
Hugoniot curve and internal energy curve

Figures 1 and 2 show GH4169 alloy's Hugoniot curve and internal energy curve, respectively. The research results of Liu et al. are consistent with the calculation results of this research, which shows that the calculation results of this paper are reliable.

Fig. 1

Hugoniot curve of GH4169 alloy

Fig. 2

Internal energy of GH4169 alloy

0K isotherm and its internal energy

Through molecular dynamics simulation of the heating process, substituting the obtained volume data into the related formula, the volume expansion coefficient αv of GH4169 alloy is 251 × 10−6K−1. After that, the 0 K isotherms of Born-Meyer potential and Morse potential and their internal energy curves are obtained by calculation. The value of Q, q, A and B are 70.55 GPa, 11.55, 151.52 GPa and 4.45, respectively.

Figures 3 and 4 show the 0K isotherm Pc and internal energy Ec of GH4169 alloy. The blue square is the result calculated using the data obtained by Liu et al. [34]. The black line represents the Born Meyer potential, and the red line represents the Morse potential. Both can be used to describe the 0 K isotherm. Figures 3 and 4 show that the 0 K isotherm Pc and its internal energy Ec both increase with the increase of V0K/V. The greater the V0K/V, the greater the growth rate of Pc and Ec. The characteristics of the cold curve are consistent with Cu and Pb. Under the condition that V0K/V is <1.5, the isotherm and internal energy curves almost coincide, which is the same as the research result of Liu et al. When V0K/V is >1.5, the isotherms and internal energy curves of the two potentials gradually separate. Whether it is isotherm or internal energy, Morse potential is closer to the results of Liu et al.'s research. This shows that when V0K/V is <1.5, both Born-Meyer potential and Morse potential are more accurate. However, when V0K/V is >1.5, the Morse potential is more accurate.

Fig. 3

Cold pressing curve of GH4169 alloy

Fig. 4

Cold energy curve of GH4169 alloy

Calculation of Grüneisen Coefficient of GH4169 Alloy
Grüneisen equation of state

Generally, Gruneisen equation of state is used to explore the behaviour of solids and to show the contribution of thermal vibration of the lattice. It's one of the most widely used equations of state in fields of high-pressure physics and explosion mechanics. Hugoniot curve is the most commonly used reference curve for the Gruneisen equation of State [28], and its classic form is: PPc=γ(V)V(EEc) P - {P_c} = {{\gamma (V)} \over V}\left( {E - {E_c}} \right)

Pc is cold press and Ec is cold energy. They are classic forms. But Hugoniot curve PH(V) and its internal energy EH(V) obtained at room temperature can also be used as a reference curve. The state equation after replacing the reference curve is expressed as PPH=γ(V)V(EEH) P - {P_H} = {{\gamma (V)} \over V}\left( {E - {E_H}} \right)

The Grüneisen coefficient is a dimensionless quantity, and the expression is γ(V)=t23V2d2[Pc(V)V2t/3]/dV2d[Pc(V)V2t/3]/dV \gamma (V) = {{t - 2} \over 3} - {V \over 2}{{{d^2}\left[ {{P_c}(V){V^{2t/3}}} \right]/d{V^2}} \over {d\left[ {{P_c}(V){V^{2t/3}}} \right]/dV}} t represents a model that characterises γ. When t = 0, Eq. (6) is regarded as Slate model γS; when t = 1, Eq. (6) refers to Dugdale-Mac Donald model γDM; when t = 2, Eq. (6) is regarded as the free volume model γ f.

When P = 0, V = V0, PS approximately satisfies Eq. (6). Therefore, Eq. (6) can also be presented as: γ(Vo)=t+23Vo2PS(Vo)PS(Vo) \gamma \left( {{V_o}} \right) = - {{t + 2} \over 3} - {{{V_o}} \over 2}{{{P^{''}}_S\left( {{V_o}} \right)} \over {P_S^\prime\left( {{V_o}} \right)}}

When t = 0, 1 and 2, there are: γ(Vo)=23Vo2PS(Vo)PS(Vo) \gamma \left( {{V_o}} \right) = - {2 \over 3} - {{{V_o}} \over 2}{{{P^{''}}_S\left( {{V_o}} \right)} \over {P_S^\prime\left( {{V_o}} \right)}} γ(Vo)=1Vo2PS(Vo)PS(Vo) \gamma \left( {{V_o}} \right) = - 1 - {{{V_o}} \over 2}{{P_S^{''}\left( {{V_o}} \right)} \over {P_S^\prime\left( {{V_o}} \right)}} γ(Vo)=43Vo2PS(Vo)PS(Vo) \gamma \left( {{V_o}} \right) = - {4 \over 3} - {{{V_o}} \over 2}{{{P^{''}}_S\left( {{V_o}} \right)} \over {{P^\prime}_S\left( {{V_o}} \right)}}

We assume that the first derivative is equal to the second derivatives of PS and PH at P = 0 and V = V0. P′H(V0) and P″H(V0) can be directly obtained by experiments or molecular dynamics simulations. Then, we can get: PS(Vo)=PH(Vo)=(CoVo)2 P_S^\prime\left( {{V_o}} \right) = P_H^\prime\left( {{V_o}} \right) = - {\left( {{{{C_o}} \over {{V_o}}}} \right)^2} PS(Vo)=PH(Vo)=4(Co2λVo3) P_S^{''}\left( {{V_o}} \right) = P_H^{''}\left( {{V_o}} \right) = 4\left( {{{C_o^2\lambda } \over {V_o^3}}} \right) PS(Vo)PS(Vo)=PH(Vo)PH(Vo)=4λVo {{{P^{''}}_S\left( {{V_o}} \right)} \over {P_S^\prime\left( {{V_o}} \right)}} = {{P_H^{''}\left( {{V_o}} \right)} \over {P_H^\prime\left( {{V_o}} \right)}} = - 4{\lambda \over {{V_o}}}

Substituting Eqs (8)–(10) into Eq. (4), we can get: γS(Vo)=2λ23 {\gamma _S}\left( {{V_o}} \right) = 2\lambda - {2 \over 3} γDM(V0)=2λ1 {\gamma _{DM}}\left( {{V_0}} \right) = 2\lambda - 1 γf(V0)=2λ43 {\gamma _f}\left( {{V_0}} \right) = 2\lambda - {4 \over 3} γS(Vo)=γDM(Vo)+13=γf(Vo)+23 {\gamma _S}\left( {{V_o}} \right) = {\gamma _{DM}}\left( {{V_o}} \right) + {1 \over 3} = {\gamma _f}\left( {{V_o}} \right) + {2 \over 3}

The Grüneisen coefficient γ is only the part of V, and γ > 0. γ is generally an empirical formula obtained by fitting experimental data. However, the Grüneisen equation of state of the complex alloy can be obtained only through numerical simulation under no experimental conditions. Therefore, the available empirical formula is as follows: γ(V)=γoVVo \gamma (V) = {\gamma _o}{V \over {{V_o}}} γ(V)=23+(γo23)VVo \gamma (V) = {2 \over 3} + \left( {{\gamma _o} - {2 \over 3}} \right){V \over {{V_o}}}

Based on the above formula, the Grüneisen equation of state of the complex alloy can be finally obtained.

Grüneisen Coefficient of GH4169 Alloy

The Ni, Cr, Fe and Mo phases are simple phases with simple crystal structures. Ni is FCC structure. Figure 5(a) shows its crystal structure. Figure 5(b) shows the BCC crystal structure of Cr, Fe and Mo. Figure 5(c) shows the Ni3Nb crystal's structure, which is the PMMN space group in the orthorhombic system. As the P63/MMC space group in the hexagonal system, Figure 5(d) presents the structure of Ni3Ti. Figure 5(e) shows the structure of Ni3Al, which is the PM-3M space group in the orthorhombic system.

Fig. 5

The crystal structure of each stable phase in GH4169 alloy

Using the PBE functions in the generalised gradient approximation (GGA), the unit cells of the seven stable phases in the GH4169 alloy are optimised. On the basis of optimising the unit cell, the energy under different volumes was calculated, and the E-V curve was obtained. Calculation of the Grüneisen coefficient γ is done by Gibbs2 software, which is based on the Quasi-Harmonic Debye model, and derived from the same formula similar to the QHA model after phonon spectrum calculation. Table 4 shows the Grüneisen coefficients of each stable phase in GH4169 calculated by Gibbs software.

Mass fraction and Grüneisen coefficient of each phase in GH4169 alloy

Phase Mass fraction (%) γ

Ni 38 2.4454
Cr 19 1.9647
γ-Fe 18 2.2604
Mo 3.50 2.2604
γ″-Ni3Nb 13.50 2.2112
γ′-Ni3Ti 4.90 2.1526
γ′-Ni3Al 3.40 2.1586

Table 5 shows the Grüneisen coefficients of GH4169 alloy obtained by fitting. By comparison, it is found that the Grüneisen coefficient obtained from the theoretical calculation proposed in this study is far from the Grüneisen coefficient obtained from the flat plate impact experiment of the first-level light gas gun. Its correctness will be verified in subsequent experiments and numerical simulations.

Grüneisen coefficient and relative error of GH4169 alloy

Grüneisen coefficient

Grüneisen coefficient in this research 2.4
Liu et al. [34] 1.82
Relative error 23.6%
Grüneisen equation of state for GH4169 alloy

Figures 6 and 7 show the 3D surface of Grüneisen equation of state.

Fig. 6

Grüneisen equation of state in P-V-E form

Fig. 7

Grüneisen equation of state in P-V-E form

Figures 6 and 7 are the equations of state respectively obtained from Eqs (18) and (19) as Grüneisen coefficients. Although viewed from the front, the two Grüneisen equations of state in P-V-E space are both concave in shape and similar in shape, but it can be found that the maximum pressures are not the same. This is caused by the difference in the Grüneisen coefficient equation. Therefore, the Grüneisen coefficient is a factor that affects the pressure value of Grüneisen equation of state.

Verification of parameter recognition

In order to further verify the correctness of this set of parameters, a plate collision experiment was carried out. By extracting the GH4169 target free surface particle speed history, the flying target and the peeling target are restored. Using the state parameter Grüneisen equation calculated in the previous section, the collision process of the plate collision experiment was numerically simulated and compared with the speed of free surface particles in the experiment. At the same time, the collision process corresponding to the speed history of the free surface particles in the numerical simulation was studied, and the morphology of the recovered flying piece and the target piece after peeling were compared. Through data comparison and morphological comparison, the accuracy of the Grüneisen state parameter equation calculated for GH4169 alloy is judged.

Principle of the experiment

The plate impact experiment is mainly used to study the high-pressure state equation of the material and to obtain parameters through the high-speed impact of flying plate on the material sample. The shock wave reaches both the free surface of the flyer and the free surface of the target from the impact surface at the same time, so that the material is in a high pressure state. When the shock wave reaches the free surface, it will produce sparse waves in the opposite direction, so that the free surface maintains the ‘zero pressure’ boundary condition. When the sparse wave from the free surface of flyer meets that from target plate, tensile stress is generated.

After the stress meets certain conditions, cracks will be generated inside the target sheet, and the crack edges will become new ‘The free side.’ When the sparse wave encounters the edge of the crack, a compression wave is emitted in the reverse direction. After the compression wave reaches the free surface, a reverse sparse wave is generated again. With the repeated generation of sparse waves and compression waves, the speed of free surface particles exhibits an oscillation form of rising and falling. It also causes the cracks to expand continuously, and finally leads to the formation of layers in the sample material.

The high-voltage device should meet two basic requirements. First, it should be able to adjust the load pressure range to obtain the Hugoniot curve from low pressure to high pressure. Second, its shock wave must have a certain degree of flatness.

In this experiment, GH4169 alloy's free surface velocity was measured by all-fibre displacement interferometer, as well as the internal state of the sample during impact, the Hugoniot elastic limit (HEL), and spallation.

Experimental Materials

When measuring the impact insulation line, the flat impact test piece needs to meet two conditions: (1) the shock wave must be a plane wave and (2) the shock wave should be uniform, that is, the parameters behind the wave front do not change with time and distance.

The experimental material is GH4169 nickel-based superalloy. It adopts a 37 mm calibre first-class light gas gun. The designed flyer diameter is 35 mm, flyer's thickness is 2 mm, the diameter of target specimen is 30 mm and the designed thickness of the target plate is 3.75 mm.

Experimental configuration

This experiment mainly adopts a symmetrical collision configuration, and the optical fibre probe and AFDISAR are used to directly measure free surface's velocity distribution on the back of the target sample.

Data processing

A total of one plate impact experiment was carried out, and the effective data of the free surface particle velocity curve was one time. The symmetrical collision configuration was adopted, the thickness of the flyer was about 2.00 mm, and the actual impact velocity range was 545 m/s. Under the condition of plate impact, the specimen is in a one-dimensional strain state, and there is circumferential hydrostatic pressure. Therefore, the relationship between the Hugoniot elastic limit (HEL) YH under the one-dimensional strain state and the yield limit Y0 of the material under the one-dimensional stress state is given by the equation YH=1v12vY0=K+4G/32GY0 {Y_H} = {{1 - v} \over {1 - 2v}}{Y_0} = {{K + 4G/3} \over {2G}}{Y_0} where ν is Poisson's ratio; K is bulk modulus and G is shear modulus. YH of the material can be determined by the speed of free surface. When free surface's velocity curve of the sample gets transition from linear deformation to nonlinear deformation, the particle velocity corresponds to the YH. When the limit free surface velocity of elastic deformation is ufm, the corresponding particle velocity (such as uHEL) can be obtained according to the law of multiplication of free surface reflection.

uHEL=12ufm {{\bf u}_{HEL}} = {1 \over 2}{{\bf u}_{fm}}

YH can be expressed as Eq. (22) by the law of conservation of momentum: YH=ρCuHEL {Y_H} = \rho {\rm{C}}{{\rm{u}}_{HEL}}

For GH4169 alloy, a total of one test was performed, and one valid data was obtained. The analysis results are shown in Table 6, and the impact velocity is 535 m/s. Figure 8 shows the free surface particle velocity results of one test. It shows that it's a ‘elastic precursor wave-plastic wave’ for the shock response of the GH4169 alloy.

Experimental results

Sample The speed of the flyer (m/s) Speed at peak (m/s) P (GPa) Ufm (m/s) uHEL (m/s) YH (GPa)
Ni-1 535 518 9.916 80 38 1.498

Fig. 8

Free surface particle velocity of GH4169 alloy

Based on the experimental curve, and using formula YH = ρCuHEL YH is calculated, as shown in Table 6.

Numerical Simulation

The nonlinear dynamics programme AUTODYN is used for simulation analysis. Set the material model with the density of GH4169 alloy as 8.24 g/cm3. The J-C constitutive model is used as the constitutive model in this paper. Tables 7 and 8 show the parameters of the state equation and constitutive equation, respectively. The parameters α and J-C constitutive parameters are quoted from Liu et al.

Grüneisen equation of state parameters of GH4169 alloy

C (m/s) S1 S2 S3 γ α
4526 1.46 0 0 2.2 0.46

Johnson-Cook constitutive model parameters of GH4169 alloy

A (MPa) B (MPa) C n m
1306 1008 0.01 0.460 1.08

The numerical analysis uses a two-dimensional plane model. A model with a 1:1 ratio of flyer to sample was established. In order to give the initial speed of the flyer, the material models of the flyer and the target were established, respectively. The parameters set in the two material models (flyer and target) are given in Tables 7 and 8, respectively. It can be seen from the experimental results that the flyer did not fail or corrode. Therefore, in order to save calculation time, only failure criteria and erosion are defined in the target model.

After setting the simulation conditions, add Gaussian point 1 (Figure 9) to the target model, and the position is at the centre of the back of the target, which is same as the position measured by AFDISAR in experiment. After that, the free surface particle velocity at the Gauss point can be obtained, which can be used to compare and analyse the velocity curve of free surface particle in the experiment, so as to verify parameters’ accuracy.

Fig. 9

Two-dimensional finite element model of GH4169 alloy plate impact test

The numerical simulation of the plate impact experiment provides the historical curve of the Gauss point 1 of the target model. Compare the curve obtained by numerical simulation with the curve obtained by experiment (Figure 10). The black curve in Figure 10 represents the speed of the free surface particles obtained through the plate collision experiment; the red curve represents the speed of the free surface particles obtained through the numerical simulation. It can be seen that the two curves have platforms at the first wave crest. Both curves have five complete waveforms in a time of 5 μm. From the last four waveforms, the particle velocity at the trough is basically the same. The velocity of the peak particles in the plate impact experiment dropped rapidly, while the velocity of the peak particles in the numerical simulation decreased slowly, causing the gap between the peak velocities to gradually increase. In addition, the disparity between the time points at which the velocity reaches the trough of the two curves keeps increasing, indicating that in the numerical simulation, the time of shock wave to propagate from free surface to spalling surface and then bounce back to the free surface is longer than in the actual plate impact experiment. Compared with the actual test results, the thickness of the slab obtained by the numerical simulation has a certain error.

Fig. 10

Free surface particle velocity curve of plate impact test and numerical simulation

The numerical simulation results of the spallation process of the plate collision experiment are shown in Figure 11. During the impact, the target was shot down. A new lamella was split on the lamina on the impact side. In the sample recovered after the plate impact experiment, flakes peeled off from the inside of the sample were also found. The experimentally recovered flyers and the non-impact side slabs have a concave centre and a conical shape as a whole. In the numerical simulation, the flyers and the non-impact side slabs also have the same deformation, which is consistent with the experimental phenomenon.

Fig. 11

Numerical simulation of spallation process

The comparison of the above data and the morphology shows that the numerical simulation results of the flyer impact experiment are consistent with experimental results. It proves that parameters of Grüneisen equation of state calculated above are consistent with the actual situation.

Conclusion

A parameter identification method of the Grüneisen equation of state for multi-element alloys is proposed. Taking GH4169 alloy as an example, parameters of Grüneisen equation of state are calculated and compared with the existing literature. When the parameters of numerical simulation and actual experiment are compared, it verified the accuracy of parameters of Grüneisen equation of state of GH4169 alloy calculated by the proposed method.

According to molecular dynamics, Hugoniot parameters C0 and λ of GH4169 alloy are calculated theoretically. The theoretical parameters are consistent with the experimental results, which prove that the theory is feasible. It's concave for Grüneisen equation of state in P-V-E space. Grüneisen coefficient model is a factor that plays a role for the maximum pressure value of the equation of state. The research shows that the calculated parameters of Grüneisen state equation of GH4169 alloy are consistent with experimental results, which primarily proves that the calculation method is accurate. Velocity history of free surface particle obtained from a flat plate impact experiment on a light air gun platform shows the morphological changes of the flyer and the specimen. Based on AUTODYN software, the calculated Grüneisen state parameter equation of GH4169 alloy was inputted into the simulation process of plate collision experiment. Comparing the experimental and simulated velocities of free-surface particle with the final morphology of the flyer and target after impact, it is concluded that the particle velocities of the free surface are basically the same, and the morphological characteristics of the flyer and the target are similar. It proves that parameters of Grüneisen equation of state obtained by the calculating method are accurate.

Although this research has achieved certain results, the application scope of the Grüneisen equation of state parameter calculation method for multi-element alloys proposed in this paper is still unclear as we have done only for GH4169 alloy. The parameter identification of Grüneisen equation of state for other metal materials needs to be explored in the future to further expand the scope of application of this method.

Fig. 1

Hugoniot curve of GH4169 alloy
Hugoniot curve of GH4169 alloy

Fig. 2

Internal energy of GH4169 alloy
Internal energy of GH4169 alloy

Fig. 3

Cold pressing curve of GH4169 alloy
Cold pressing curve of GH4169 alloy

Fig. 4

Cold energy curve of GH4169 alloy
Cold energy curve of GH4169 alloy

Fig. 5

The crystal structure of each stable phase in GH4169 alloy
The crystal structure of each stable phase in GH4169 alloy

Fig. 6

Grüneisen equation of state in P-V-E form
Grüneisen equation of state in P-V-E form

Fig. 7

Grüneisen equation of state in P-V-E form
Grüneisen equation of state in P-V-E form

Fig. 8

Free surface particle velocity of GH4169 alloy
Free surface particle velocity of GH4169 alloy

Fig. 9

Two-dimensional finite element model of GH4169 alloy plate impact test
Two-dimensional finite element model of GH4169 alloy plate impact test

Fig. 10

Free surface particle velocity curve of plate impact test and numerical simulation
Free surface particle velocity curve of plate impact test and numerical simulation

Fig. 11

Numerical simulation of spallation process
Numerical simulation of spallation process

Grüneisen coefficient and relative error of GH4169 alloy

Grüneisen coefficient

Grüneisen coefficient in this research 2.4
Liu et al. [34] 1.82
Relative error 23.6%

Grüneisen equation of state parameters of GH4169 alloy

C (m/s) S1 S2 S3 γ α
4526 1.46 0 0 2.2 0.46

Mass fraction and Hugoniot parameters of each phase in GH4169 alloy

Phase Mass fraction (%) C0 λ

Ni 35 4.4133 1.1419
Cr 20 4.9712 1.1196
γ-Fe 19 3.8327 2.2667
Mo 5.00 5.9806 0.9444
γ″-Ni3Nb 12.60 5.5252 2.6675
γ″-Ni3Ti 4.80 5.6461 1.0397
γ″-Ni3Al 3.60 5.3751 1.068

Mass fraction and Grüneisen coefficient of each phase in GH4169 alloy

Phase Mass fraction (%) γ

Ni 38 2.4454
Cr 19 1.9647
γ-Fe 18 2.2604
Mo 3.50 2.2604
γ″-Ni3Nb 13.50 2.2112
γ′-Ni3Ti 4.90 2.1526
γ′-Ni3Al 3.40 2.1586

Hugoniot parameters and relative error of GH4169 alloy

C0 λ

Hugoniot parameters in this research 4.532 1.62
Liu et al. [34] 4.485 1.45
Relative error 0.70% 15.60%

Mass fraction and atomic fraction of each element of GH4169 alloy

Element Mass fraction (%) Atomic fraction (%)

Ni 53.40 52.40
Cr 18 20.80
Fe 18.20 19.50
Mo 3.50 1.40
Nb 4.80 2.60
Ti 1.50 1.90
Al 0.60 1.40

Johnson-Cook constitutive model parameters of GH4169 alloy

A (MPa) B (MPa) C n m
1306 1008 0.01 0.460 1.08

Experimental results

Sample The speed of the flyer (m/s) Speed at peak (m/s) P (GPa) Ufm (m/s) uHEL (m/s) YH (GPa)
Ni-1 535 518 9.916 80 38 1.498

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