LIU Haowei, SU Buyun, QIU Ji, LI Zhiqiang. Numerical Simulation of Multiaxial Creep Behavior of 2D Anisotropic Cellular Materials[J]. Chinese Journal of High Pressure Physics, 2020, 34(6): 064202. doi: 10.11858/gywlxb.20200561
Citation:
ZHANG Ruike, GUO Rui’ang, XIAO Xiong, HE Duanwei. Measurement of the Melting Point of Hexagonal Boron Nitride under Pressures below 5 GPa[J]. Chinese Journal of High Pressure Physics, 2024, 38(6): 063401. doi: 10.11858/gywlxb.20240813
LIU Haowei, SU Buyun, QIU Ji, LI Zhiqiang. Numerical Simulation of Multiaxial Creep Behavior of 2D Anisotropic Cellular Materials[J]. Chinese Journal of High Pressure Physics, 2020, 34(6): 064202. doi: 10.11858/gywlxb.20200561
Citation:
ZHANG Ruike, GUO Rui’ang, XIAO Xiong, HE Duanwei. Measurement of the Melting Point of Hexagonal Boron Nitride under Pressures below 5 GPa[J]. Chinese Journal of High Pressure Physics, 2024, 38(6): 063401. doi: 10.11858/gywlxb.20240813
In this study, a flash-heating device was designed and assembled to achieve instantaneous discharge heating of samples under high pressure in the DS 6×14 MN domestic hinged six-anvil large chamber press. By combining large chamber static high pressure and instantaneous discharge heating technologies, the melting state of crystals has been determined by the nucleation and growth characteristics during solidification. The melting behavior of h-BN powder crystal under high pressure was studied by instantaneous discharge heating treatment. Using scanning electron microscopy (SEM), the microstructures of samples obtained by high-pressure instantaneous discharge heating treatment was analyzed in order to assess the melting state of h-BN crystals. It was determined that the melting points of h-BN under 3.4 and 4.3 GPa are (4251±150) K and (4531±200) K, respectively. These results are beneficial for exploring the applications of h-BN and revising the existed temperature-pressure phase diagram of boron nitride.
The corrugated interface between different fluids grow when accelerated from a low-density fluid to a high-density fluid, which is called Rayleigh-Taylor (RT) instability[1-2].This phenomenon may also occur in metals, but usually under a high pressure and at a high strain-rate, but differs most distinctly from the RT instability in fluids in its strength effect of the metal, which can stabilize the perturbation growth[3-4] and make the metallic RT instability more complex and difficult.Here, it is also affected by the loading state and the properties of the metallic materials.The metallic RT instability at high pressure and strain-rate can be observed in inertial confinement fusion[5], supernova explosion[6], asteroid collision[7], the motion of earth's inner core and plate tectonics[8], and so on.Therefore, the metallic RT instability is currently a major concern for researchers and receives a great deal of academic attention.
In theoretical studies about the metallic interface instability, dispersion relations of the perturbation growth are derived mainly based on the energy[9-11] or force equilibrium[12-13].However, the previous linear analysis cannot predict the perturbation growth accurately just by applying the perfect plastic constitutive relation and constant pressure loading state.Based on the energy balance, a perturbation growth equation using Steinberg-Guinan (SG) and the Johnson-Cook constitutive models, as well as a variable pressure loading process consistent with experiments, has been derived that precisely predicts the growth of metallic RT instabilities driven by detonation and laser plasma.However, the linear analysis still has its limitations and does not take full account of the loading procedure.
Experimental studies of the metallic RT instability started in the 1970s.The pioneering experimental research[14] was the perturbation growth of a flat aluminum plate accelerated by the expansion of detonation products, which was observed using a high-energy X-ray facility.What was achieved then inspired researchers, and the similar equipment was utilized in later research[15-17].In the USA and Russia particularly, numerous numerical simulations and experimental investigations for the metallic RT instability have been carried out, but have mainly concentrated on the perturbation growth and such influencing factors as the initial amplitude, the wavelength and material properties.Igonin and Ignatova et al.[18-19] experimentally and numerically studied the dynamic behaviors of copper (Cu) and tantalum (Ta) subjected to both shock and shockless loading by employing a perturbation growth method.They observed that the formation of the bi-periodic twin structures resulted in an initial loss of the shear strength of Cu, but failed to observe localization in Ta.Olson et al.[16] experimentally studied the effects of the grain size and material processing on the RT perturbation growth of Cu.They found that both the single-crystal orientation and the strain hardening due to the material processing can affect the perturbation growth, but the polycrystalline grain size cannot.For the plane detonation, the loading pressure is generally about 30 GPa.To enhance the loading pressure, Henry de Frahan et al.[17] studied the beryllium RT instability using an iron flyer plate to impact the second high explosive (HE) to raise the pressure to 50 GPa in their experiments, and combined numerical simulations to calibrate the feasibility of different constitutive models.When the sample is driven by electromagnetism[20-21] or laser[4, 22], the loading pressure can be further increased.Very extreme conditions of pressures over 1 000 GPa and strain rates of 108 s-1 have been achieved at the National Ignition Facility, USA, where the RT instability experiment in vanadium was carried out, and constitutive models in solid phase were tested by comparing simulations with experiments measuring the perturbation growth[23] under the extreme conditions mentioned.
In the metallic interface instability, the perturbation growth is related to and arrested by the material strength.Moreover, some investigations have demonstrated that the material strength increases under these extreme conditions.Results from the metallic RT experiments and computations by Barnes et al.[14] show that the yield strength of 1100-0 aluminum is over 10 times larger than the standard parameter, and the yield strength of 304 stainless steel also increases by more than three times.Using the SG constitutive strength model, calculations of plasma-driven quasi-isentropic RT experiments of Al-6061-T6 using the Omega laser at a peak drive pressure of 20 GPa indicate that its yield strength is a factor of about 3.6 times over the ambient value[22].In Park et al.'s[4] plasma-driven quasi-isentropic polycrystalline vanadium RT experiments using the Omega laser with a peak drive pressure of 100 GPa, the measured RT growth was substantially lower than predictions using the existing constitutive models (SG and Preston-Tonks-Wallace) that work well at low pressures and long timescales.Using the SG model, the simulations agree with the RT experimental data when the initial strength is raised by a factor of 2.3.Therefore, the SG and Preston-Tonks-Wallace models underestimate the strength of vanadium under very high pressures and strain rates.Belof et al.[24] first measured the dynamic strength of iron undergoing solid-solid phase transition by using RT instability.In conjunction with detailed hydrodynamic simulations, the analysis results revealed significant strength enhancement of the dynamically generated ε-Fe and reverted α′-Fe, comparable in magnitude to the strength of austenitic stainless steels.Therefore, the metallic RT instability was suggested and used as a tool for evaluating the material strength of solids at high pressures and high strain rates[3, 25], and then for modifying or developing new constitutive models for these conditions[26-27].
In view of the dominant role of the material strength in metallic interface instabilities, and the limitations of existing constitutive models at high pressures and high strain rates, we aimed to investigate the material strength and its effects on metallic interface instabilities.In this paper, we also conducted an RT instability experiment in explosion-driven aluminum, and measured the perturbation growth using X-ray radiography.In combination with elastic-plastic hydrodynamic simulations, we investigated the dynamic behavior of metallic RT instabilities and the role of the material strength in these.
1.
Experimental Setup
Following that of Barnes et al., [14]our experiment used the setup as shown in Fig. 1(a), where we have a sketch of the experimental setup consisting of a detonator, a booster, plane wave lens, JO-9159 HE (100 mm in diameter and 50 mm in thickness), an aluminum sample, and a vacuum.Fig. 1(b) shows the experimental sample of aluminum with a diameter of 65 mm and a thickness of 1.5 mm in the central region.An initial sinusoidal perturbation was machined on the side of the aluminum sample facing the HE.The perturbation amplitude and wavelength were 0.25 mm and 6 mm, respectively.The HE products crossed the void of 3.5 mm between the sample and HE and accumulated on the perturbation interface of the sample, providing a smooth rise to peak pressure and a quasi-isentropic drive.Moreover, the void between the sample and HE can ensure that the temperature of the sample at high pressures remain below the melting point[22].
Figure
1.
Sketch of the experimental setup and sample
In the experiment, X-ray radiography was used to record the evolution of the perturbation interface from the JO-9159 explosive detonation at zero time.A Doppler pin system was used to measure the history of the free surface velocity, which can be integrated to obtain the corresponding free surface displacement.Fig. 1(c) shows the distribution of measurement points of the free surface velocity, where both points 1 and 2 correspond to the wave trough positions with one wavelength interval, and point 3 corresponds to the wave crest position with 1.5 wavelength intervals from point 2.
2.
Numerical Methods
Based on our indoor hydrodynamic code of compressible multi-viscous flow and turbulence (MVFT)[28-30], we developed a detonation and shock dynamics code with high fidelity by considering the explosive detonation and the elastic-plastic behavior of the material.This code can be used to study the physical problem of multi-materials, large deformation, and strong shock.The governing equations in conserved form are as follows
where i and j represent the x, y, and z directions; V is the control volume, S the surface of control volume, n the unit vector of the external normal direction, ρ, uk (where k=i, j), p, and E are the density, velocity, pressure, and total energy of per unit mass; and sij the deviation stress tensor.
The physical problem as described by Eq.(1) was decomposed into three one-dimensional problems.For each of them, the physical quantities in a cell were interpolated and reconstructed using a piecewise parabolic method (PPM).The one-dimensional problem was then resolved using a two-step Euler algorithm:First the physical quantities were solved by the Lagrange matching, and then remapped back to the stationary Euler meshes.The effect of material strength, explosive detonation, and artificial viscosity were implemented in the Lagrange step.The multi-material interface was captured by applying a volume-of-fluid (VOF) method.
2.1
Equation of State
In our numerical simulations, the equation of state (EOS) for the explosive and aluminum are the Jones-Wilkins-Lee (JWL) and Mie-Grüneisen equations of state, respectively.The Jones-Wilkins-Lee equation of state is
p(ρ,T)=A(1−ωR1v)e−R1v+B(1−ωR2v)e−R2v+ωˉEv
(2)
where v=ρ0/ρ is specific volume; A, B, R1, R2, and ω are constants; and E is the internal energy per unit volume.Table 1 lists the JWL EOS parameters of the JO-9159 explosive.The Mie-Grüneisen equation of state is
Table
1.
Equation of state parameters of JO-9159 explosive
where μ=ρ/ρ0-1 is the relative compression, ρ0 the initial density, c the sound velocity at zero pressure, γ0 the Grüneisen coefficient, and a, S1, S2, and S3 are constants (in Table 2).
Table
2.
Mie-Grüneisen equation of state parameters of aluminum
In our simulations, the elastic-plastic behavior of aluminum at high pressures and high strain rates was described using the SG constitutive model.The SG model introduces pressure, temperature, and strain-rate terms into the elastic-plastic constitutive equation, while the coupling effect of pressure and strain rate on flow stress was characterized by the separating variables.Additionally, as the flow stress in the SG model relies on pressure, there is a coupling relationship between the material constitutive equation and the equation of state, which indicates the feature of pressure hardening of metal under high pressure.The dynamic yield strength YSG and the shear modulus G determined by the SG model are expressed as
YSG=Y0[1+β(εp+εi)]n[1+Apη−1/3−B(T−300)]
(4)
G=G0[1+Apη−1/3−B(T−300)]
(5)
where Y0 and G0 are the initial yield strength and the shear modulus, respectively; β and n are the material strain hardening coefficient and the hardening index, respectively; A is the pressure hardening coefficient; η=ρ/ρ0 is the material compression ratio; and B is the temperature softening coefficient (in Table 3).
Table
3.
Steinberg-Guinan constitutive model parameters of aluminum
In our experiment, X-ray radiography recorded an image of the perturbed interface at 7.78 μs, as shown in Fig. 2(a), from which we obtained the amplitude of 0.77 mm simultaneously by image processing.In the simulations, the mesh size was 15.6 μm, and Tables 2 and 3 list the parameters of the Mie-Grüneisen EOS and the SG constitutive model of aluminum, respectively.
Figure
2.
Comparisons of the perturbed interface between experiment and numerical simulations ((a) Experimental image, (b) Simulated image at normal strengths Y0 and G0, (c) Simulated image at 10 times the normal strengths Y0 and G0)
Fig. 3 shows the pressure histories of the crest (solid line) and the trough (dashed line) on the loading surface, which increase continuously and smoothly in a short time and form an approximate quasi-isentropic drive.Afterwards, the expansion of the detonation products decelerates gradually, and the loading pressure on the interface rises slowly.However, the pressure at the trough ascends faster than that at the crest, and the peak pressure at the trough is also relatively larger, because the detonation products converges at the trough and diverges at the crest.The average peak pressure is about 25 GPa, and the strain rate reaches 106 s-1.The loading pressure then reduces gradually, which is attributed to the decrease of the expansion pressure of the detonation products and the unloading effect of the rarefaction wave reflecting from the free surface.
Figure
3.
Pressure histories of crest and trough at the loading surface
Fig. 4 shows several contours of local pressure (a), density (b), and temperature (c) at 6 different times after the arrival of the detonation products at the loading surface, from left to right and top to bottom, at 6.36, 6.5, 6.7, 6.9, 7.1 and 7.3 μs.They exhibit an evolution process of the perturbed interface and the interaction of the wave and the interface.The blue part of the temperature contour is the aluminum sample, and the sample temperature remains below 500 K and far below the melting point of 1 200 K, which indicates that the sample is in the elastic-plastic state all the time.
Figure
4.
Contours of local pressure (a), density (b), and temperature (c) at 6.36, 6.5, 6.7, 6.9, 7.1, and 7.3 μs from left to right and top to bottom after the arrival of detonation products at the loading surface
Fig. 2(b) shows an image of the sample at 7.78 μs when the initial yield strength Y0 and the shear modulus G0 are normal values.Fig. 5 shows a comparison of the perturbation amplitudes between the experiment and numerical simulations, where the square symbol corresponds to the experimental result, and the solid black line corresponds to the numerical results when Y0 and G0 are normal.The simulated amplitude is much larger than that in the experiment when using normal values of Y0 and G0.This is because the aluminum strengthens under such conditions, and the SG constitutive model underestimates its strength, which can suppress the perturbation growth.
Figure
5.
Growth histories of the perturbation amplitude
Fig. 6 shows the time histories of the free surface velocity (a) and displacement (b) at 3 measurement points (dot-dot-dashed lines:experiment; solid, dashed, and dotted lines:simulations), which agree well with each other.Therefore, the calculations of detonation of the explosive and the thermodynamic state of the sample are exact.
Figure
6.
Time histories of the free surface velocity (a) and displacement (b)
Fig. 7 shows the calculated time histories of the strain at the crest (solid line) and trough (dashed line) of the loading surface.The deformation at the trough is much larger than that at the crest because the trough of the sample is in a stronger tensile stress state, which is the main mechanism for the deformation of the perturbation interface.Fig. 8 shows the time histories of the dynamic yield strength at the crest (solid line) and trough (dashed line) of the loading surface, calculated using the SG constitutive model, similar to the profile of the loading pressure, which demonstrate that the material strength increased as did the loading pressure under a certain condition.
Figure
7.
Time histories of strain at the crest and trough of the loading surface
Moreover, when the loading pressure reaches a peak, the dynamic yield strength was up to 3 times that of the initial value.In fact, the normal SG model is generally calibrated by conventional Hopkinson and Taylor impacts experiment with a lower strain rate.Under the current loading conditions (loading pressure of 25 GPa and strain rate of 106 s-1), the strength is not great enough to suppress the perturbation growth.However, when the initial yield strength Y0 and the shear modulus G0 increase to 10 times that of the normal values, good agreement between the experiment and simulation is achieved, as shown in Fig. 2(c) for the perturbation interface and in Fig. 5 for the perturbation amplitude with the dashed line.Therefore, the material strength intensively stabilizes the perturbation growth.The dotted lines in Fig. 5 are fitted lines from the simulated results, indicating that the perturbation amplitude grows exponentially over time.
We studied the effect of the initial yield strength and the initial shear modulus of the material on the evolution and growth of the perturbed interface.Figs. 9(a), 10(a), and 11(a) show the calculated growth histories of perturbation amplitude, strain histories, and dynamic yield strength histories at the trough of the loading surface, respectively, when the initial yield strength is fixed at the normal value and as the initial shear modulus increases gradually.The growth of the perturbation amplitude exhibits no change even when the initial shear modulus increases to 10 times that of the normal value, which means that the initial shear modulus has no influence on the material deformation and does not affect the dynamic yield strength.
Figure
9.
Growth histories of the perturbation amplitude for different values of initial shear modulus (a) and yield strength (b)
Figure
11.
Time histories of yield strength at the trough of the loading surface for different values of initial shear modulus (a) and yield strength (b)
Figs. 9(b), 10(b), and 11(b) show the numerical results when the initial shear modulus is fixed and the initial yield strength gradually increases to 10 times that of the normal value.These indicate that, with the increase of the initial yield strength, the dynamic yield strength also increases, the material deformation is retarded, and the perturbation growth is suppressed markedly.Therefore, the initial shear modulus of the material exerts no effect on the growth of the metallic RT instability within a certain range, while the initial yield strength has an obvious effect on it.
4.
Conclusions
We have established an experimental setup and developed a numerical simulation method to investigate the RT instability in metallic materials driven by explosion.We also studied the RT instability in aluminum, and drew the following conclusions:
(1) The perturbation amplitude grows following an exponential law over time.
(2) When using the normal physical property parameters of aluminum, simulated evolution of the perturbed interface agrees with experiment only qualitatively, and there is a big quantitative difference between them because the aluminum strengthens under high pressures and at high strain rates, and the SG constitutive model underestimates its strength as being not great enough to suppress the perturbation growth.
(3) When the initial yield strength and the initial shear modulus increase to 10 times their normal values, the numerical and experimental results are in good agreement both qualitatively and quantitatively.The underlying physical mechanism is the stabilization effect of material strength on the perturbation growth.Moreover, the initial shear modulus has no influence on the perturbation growth within a certain range whereas the initial yield strength does influence it strongly.Therefore, the material strength dominates the evolution of the metallic RT instability.
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LIU Haowei, SU Buyun, QIU Ji, LI Zhiqiang. Numerical Simulation of Multiaxial Creep Behavior of 2D Anisotropic Cellular Materials[J]. Chinese Journal of High Pressure Physics, 2020, 34(6): 064202. doi: 10.11858/gywlxb.20200561
LIU Haowei, SU Buyun, QIU Ji, LI Zhiqiang. Numerical Simulation of Multiaxial Creep Behavior of 2D Anisotropic Cellular Materials[J]. Chinese Journal of High Pressure Physics, 2020, 34(6): 064202. doi: 10.11858/gywlxb.20200561
Figure 1. Schematic diagrams of the internal circuit (a) and connection assembly (b) of the flash-heating device (The capacitors are arranged in parallel by using 10 AWG thick wires and divided into upper-level and lower-level. The maximum output voltage is 1 kV. Simultaneous charging and discharging are not permissible.)
Figure 2. SEM images of NaCl samples after flash-heating with different voltages under (3.4±0.2) GPa (The red circled areas indicate the locations where fine grains are observed, with a distinct grain refining at 300 V. Multiple experiments are conducted, sequentially reducing the voltage to 70, 35, and 30 V, all of which resulted in the appearance of fine grains, demonstrating that at such voltages, the temperature is above the melting point of NaCl. When the voltage reduced to 25 V, no fine-grained areas can be identified in the SEM image; instead, only fragmented grains due to pressure are observed. This suggests that the melting voltage of NaCl under 3.4 GPa is between 25 and 30 V, approximately (27.5±2.5) V. At this pressure, the melting point of NaCl is 1582 K, suggesting that a voltage of 27.5 V induces an instantaneous temperature of approximately 1582 K in the sample chamber.)
Figure 3. SEM images of KCl samples after flash-heating under 3.4 GPa (No grain refinement was observed at 21 V, but appeared at 25 V within the red-circled areas. It indicates that the melting voltage of KCl under 3.4 GPa is between 21 and 25 V, approximately (23±2) V. At this pressure, the melting point of KCl is 1416 K, suggesting that a voltage of 23 V induces an instantaneous temperature of approximately 1416 K in the sample chamber.)
Figure 4. SEM images of Al2O3 samples after flash-heating under 3.4 GPa (No grain refinement was observed at 46 V, but appeared at 51 V within the red-circled areas. It indicates that the melting voltage of Al2O3 under 3.4 GPa is between 46 and 51 V, approximately (48.5±2.5) V. At this pressure, the melting point of Al2O3 is 2701 K, suggesting that a voltage of 48.5 V induces an instantaneous temperature of approximately 2701 K in the sample chamber.)
Figure 5. SEM images of MgO samples after flash-heating under 3.4 GPa (No grain refinement was observed at 65 V, but appeared at 70 V within the red-circled areas. It indicates that the melting voltage of MgO under 3.4 GPa is between 65 and 70 V, approximately (67.5±2.5) V. At this pressure, the melting point of MgO is 3419 K, suggesting that a voltage of 67.5 V induces an instantaneous temperature of approximately 3419 K in the sample chamber.)
Figure 6. Flash-heating voltage vs. temperature relationship for the flash-heating device (Black dots represent the temperature calibration points by four standard materials, and blue dots are experimental data from Liang, et al.[15])
Figure 7. SEM images of h-BN samples after flash-heating at different voltages under 3.4 GPa (The red circled areas indicate the locations where fine grains are observed, with a distinct grain refining at 300 V. Grain refinement is still observable at 96 V, but cannot be observed at 90 V. It indicates that the melting voltage of h-BN under this pressure is approximately (93±3) V. The corresponding melting temperature is approximately (4251±150) K.)
Figure 8. SEM images of h-BN samples after flash-heating at 4.3 GPa (No grain refinement is observed at 97 V, but appeared at 105 V within the red-circled areas. It suggests that the melting voltage of h-BN under this pressure is approximately (101±4) V. The corresponding melting temperature is approximately (4531±200) K.)
Figure 9. Extension and comparison of the results obtained in this study with data of Zhang et al.[16] on the melting curve of c-BN (The blue region indicates the possible range of the c-BN melting curve, the green region represents the possible range of the h-BN melting curve obtained from this study, and the middle red region indicates the possible range of the triple point. The melting point of h-BN is expected to increase with pressure, with the slope gradually steeper but not negative. This differs from the predictions made by Solozhenko et al.[14].)
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