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We conducted numerical simulations of the related processes of interface instability, tensile fragmentation, and jetting resulting from four kinds of typical macro defect perturbations (chevron, sine wave, rectangle, and square) on a Cu free surface under a reflected shock wave when Cu impacts a solid wall at a speed of 2.5 km/s and found that, for the chevron and sine wave cases, the ejecta velocities of the head are 6.28 and 5.88 km/s, respectively. Some parts of the inner material are in a tensile state without any fragmentation, which is observed only in the main body of the material owing to the tension effect. Furthermore, for the other two initial perturbations (rectangle and square), the highest ejecta velocities may even reach 9.14 and 9.59 km/s, respectively. Fragmentation caused by multilayer spallation can be observed on a large scale in the Cu main body, and there are granules in the front area of the ejecta but the degree to which fragmentation occurs is much less in the Cu main body and there is a notable high-speed, low-density granule area in the ejecta head. Finally, we present a detailed analysis of the spatial distribution of the granules, ejecta mass, pressure, temperature, and grid convergence.

Metal interface instability occurs as a reaction to the effect of a shock wave or acceleration or to a shear load on the perturbed interface, and may afterward lead to fragmentation and mixing. This phenomenon, common in explosions and shock processes under extreme conditions (high temperature and high pressure), is jointly controlled by three scale factors, i.e., 1) the thermodynamics and the geometric boundary (macroscopic scale), 2) the initial perturbation of the metal interface (mesoscopic scale), and 3) material properties (microscopic- mesoscopic scale). Metal interface instability is a typical multiscale, multiphysical, strongly coupled, nonequilibrium, and complex flow phenomena.

Related studies of the Rayleigh-Taylor (RT) instability of metal began in the 1970s. In particular, in the USA and Russia, theoretical studies, numerical simulations, and experimental investigations have been conducted. The prior research by Barnes, Blewett, McQueen, Meyer, and Venable [

The theoretical study of the metal RT instability is limited by the material models used and also the lack of comparison with related experiments. Particularly, at the turbulent mixing stage, it is difficult to predict the development of the interface instability. Despite some progress made in experimental studies, understanding of metal instability in detail is still hindered by diagnostic techniques. Therefore, because the problems in algorithm precision, the constitutive model, and the equation of state (EOS) are still too tough to solve, simulation work stands out as especially worthwhile. Some effective software and codes, such as ABQUS, Lagrange codes FCI2, MAGEE, TOODY II [

Under shock loading, macro defects perturbations on metal free surface may lead to ejecta and fragmentation. We numerically studied the growth of the instability on the metal interface with four kinds of typical macro defect (chevron, sine wave, rectangle, and square) perturbations and the dynamic characteristics of ejecta.

Given a multimaterial, large-deformation, strong-shock physics problem, we improved our previous large eddy simulation code MVFT (multiviscous flow and turbulence) [

∂ ∂ t ∫ V ρ d V = − ∮ S ρ u i n i d S , ∂ ∂ t ∫ V ρ u j d V = − ∮ S P n j d S + ∮ S s i j n i d S − ∮ S ρ u i u j n i d S , ∂ ∂ t ∫ V ρ E d V = − ∮ S u j P n j d S + ∮ S u i s i j u j d S − ∮ S ρ u j E n j d S , (1)

where V is the control volume, u_{j} is the velocity, S is the surface area of control volume, n → is the external normal line, s_{ij} is the deviatoric stress tensor, P is the static pressure, and E is the total energy per unit mass.

First, in our numerical simulation, a dimension splitting method is used to split Equation (1) into three one-dimensional problems, which are then solved using the piecewise parabolic method (PPM) method to perform the interpolation and reconstruction of the physical quantity in each grid. Owing to the lack of automatic monotonicity of PPM, there would be numerical oscillation at the discontinuity point, which leads to a decrease in the accuracy of the discontinuous solution. To restrict the numerical oscillation, we introduce a flow restrictor. In adopting this method, a monotonic limiter is utilized so that the calculation of δQ_{j} is limited monotonically, as shown in the following:

to keep Q j + 1 / 2 n between Q j n and Q j + 1 n , where the Q are conservation quantities.

In a Lagrange step, the material strength model, artificial viscosity, and explosive detonation model are needed for calculation and, therefore, the Jones-Wilkins -Lee (JWL) equation of state is used for the explosive detonation and the Mie-Gruneisen equation of state is applied for the strength material. As for the strength model, we utilize the Steinberg-Guinan constitutive model.

In the Steinberg-Guinan constitutive model [

G ( P , T ) = G 0 [ 1 + 1 G 0 ( ∂ G ∂ P ) 0 η − 1 / 3 P + 1 G 0 ( ∂ G ∂ T ) 0 ( T − 300 ) ] , (3)

σ SG = Y 0 ( 1 + B ε ) n [ 1 + A η − 1 / 3 P − α ( T − 300 ) ] , (4)

where Y_{0} signifies the yield strength in the initial state, G_{0} is the shear modulus in the initial state, P and T are the pressure and temperature, ( ∂ G / ∂ P ) 0 and ( ∂ G / ∂ T ) 0 represent, respectively, the partial derivatives of shear modulus to the pressure and temperature in the initial state, A and α correspond to ( ∂ G / ∂ P ) 0 / G 0 and ( ∂ G / ∂ T ) 0 / G 0 , B and n are material strain hardening parameters, e is the strain, and η = ρ / ρ 0 is the material compression ratio.

The form taken by the Steinberg-Guinan constitutive model is irrelevant to the strain rate, yet the requirement upon the strain rate is that it should be >10^{5} s^{−1}. The reason for this restriction is that the effect of metal softening counteracts that of metal hardening, so this model can be used to describe multimaterial flow stress under high pressure, which is also the mostly used constitutive model under high pressure.

Through simulation of Barnes’s three experiments [

In the first experiment (wavelength 5.08 cm and amplitude 0.02 cm), it can be seen that, by a seminumerical linear analysis, when Y_{0} = 0.055 GPa, the numerical simulation agreed well with the experiment data. Although the 2D calculations of our program agree well with the experiment (Y_{0} = 0.075 GPa), there are differences between the above results and those from the 1100-0 Al Steinberg- Guinan constitutive model under the condition of Y_{0} = 0.04 GPa. This is true especially when Y_{0} = (0.325 GPa) and (0.2 GPa), in which case the results significantly differ from those in ref. [

ρ 0 (g/cm^{3})^{ } | A (Mbar) | B (Mbar) | R 1 | R 2 | ω | E 0 (Mbar) | D c j (km/s) |
---|---|---|---|---|---|---|---|

1.84 | 8.524 | 0.1802 | 4.6 | 1.3 | 0.38 | 0.102 | 8.8 |

ρ 0 (g/cm^{3})^{ } | c (km/s) | γ 0 (Mbar) | a | S_{1} | S_{2} | S_{3} |
---|---|---|---|---|---|---|

2.707 | 5.25 | 1.97 | 0.47 | 1.37 | 0.0 | 0.0 |

G 0 (GPa) | Y 0 (GPa) | Y max (GPa) | B | n | A (GPa^{−}^{1}) | α (kK^{−}^{1}) |
---|---|---|---|---|---|---|

27.1 | 0.04 | 0.48 | 400.0 | 0.27 | 0.0652 | 0.616 |

It is easily observed from _{0} declines to 0.04 GPa, the growth of the amplitude calculated by our program is greater than that in the experiment, which indicates that the influence of the strength on the amplitude growth is weak. For the case when Y_{0} = 0.075 GPa, the simulation result agrees well with that from the experiment. However, the result for Y_{0} = 0.055 GPa coincides with Barnes’s further analytical results, which took account of the deviation of X-ray photographic magnification [

For the other two experiments, we conducted the simulation based on the initial yield strength Y_{0} = 0.075 GPa of 1100-0 Al calibrated by using our program. In the simulation of the experiment (with wavelength 2.54 cm and amplitude 0.01 cm), three kinds of grid resolution were adopted: Dx = Dy = 4, 8, and 15 μm. In

In summary, when the experimental conditions are P ≈ 10 GPa and ε ˙ ≤ 10 5 s − 1 , our calculation is in accordance with the experiment, and the grid convergence is also in the same trend.

Configuration of the calculation modes are shown in ^{2}, and the Cu mass of defect unit thickness is 2.1 g. For the chevron, the defect shape is an isosceles right-angled triangle with a 1-cm-long bottom edge. The sine wave amplitude and wavelength are 0.3927 and 2 cm, respectively. Additionally, the length and width of the rectangle are 0.7071 and 0.35355 cm, and the length of the square is 0.5 cm.

To verify grid convergence, we take the chevron model as an example and select two different sizes (25 and 12.5 µm).

ρ 0 (g/cm^{3}) | c (km/s) | γ 0 (Mbar) | a | S_{1} | S_{2} | S_{3} |
---|---|---|---|---|---|---|

8.93 | 3.94 | 1.99 | 0.47 | 1.489 | 0.0 | 0.0 |

G 0 (GPa) | Y 0 (GPa) | Y max (GPa) | B | n | A (GPa^{−1}) | α (kK^{−}^{1}) |
---|---|---|---|---|---|---|

47.7 | 0.12 | 0.64 | 36.0 | 0.45 | 0.0283 | 0.377 |

condition are 0.64% and 0.15%, respectively. In

The left panel of

large fragments is almost the same in the two different grids, and that of the smallest fragment even reaches one grid size, so the size of the smallest fragment is linked to the computational grid. The right panel of

The above findings reveal that the numerical calculations of the fragmentation are related to the size of the calculation grid. To eliminate the influence of the grid’s mesh, we chose the same grid size, 12.5 µm in length, in the four models. Thus, the Euler mesh encompasses 12,800,000 points in the whole computational domain [0 cm, 10 cm] ´ [−1 cm, 1 cm].

To compare the various characteristics of the interface instability, fragmentation, and ejecta under the influence of the four typical defects,

Time (µs) | Ejecta mass (g) | |||
---|---|---|---|---|

Chevron | Sine wave | Rectangle | Square | |

10.5 | 0.533 | 0.581 | 0.662 | 0.586 |

13.5 | 0.737 | 0.803 | 0.848 | 0.712 |

0.803, 0.848, and 0.712 g, respectively. In summary, although all the ejecta masses increase with time, it is obvious that the law of ejecta mass and growth varies according to the kind of typical macro defect perturbation. The left panel of

For further analyze of the velocity and spatial distributions of the jet particles,

low-density area can be found only in part of the ejecta head, indicating that there is tensile stress in the material, without any fragmentation. This explains why no discrete broken particles are found in the front of the ejecta and why only fragmentation can be found in the main body owing to the stretch. However, the ejecta velocities for the rectangle and square are even higher than in the former two conditions, and the highest velocities reach 9.14 and 9.59 km/s, respectively. Additionally, the discrete density distributions are distinctly presented in

and 8.16 - 8.27 cm. For the square, the particle distribution domains are 7.61 - 8.30, 8.92 - 9.19, 9.33 - 9.44, and 9.59 - 9.82 cm, so the conclusion may be drawn that the square undergoes more serious fragmentation than the rectangle. On a macro level, both ejecta heads of the square and rectangle are in a high-speed, low-density area in the particle state. At the same time, the fragmentations and the distribution are closely connected with the initial defect shape. To further understand the particles’ thermodynamic state,

By considering four kinds of typical macro defect (chevron, sine wave, rectangle, and square) perturbations on a Cu free surface, our research in the present work

mainly focuses on the interface instability, fragmentation, and jetting under a reflected shock wave. Through numerical simulation, we quantitatively compared the different defects and concluded that the interface instability, fragmentation, and ejecta all originate from the initial interface defect and are also associated closely with the shape of the defect. According to the above findings, in the chevron and sine wave cases, the ejecta mass velocities of the head are 6.28 and 5.88 km/s, respectively. Some parts of the inner material are found in a tensile state without any fragmentation, which appears only in the main body of the metal owing to the tension effect. Additionally, for the other two initial perturbations (rectangle and square), the highest ejecta mass velocities are 9.14 and 9.59 km/s, respectively. Fragmentation appears in the large range of shaped pole owing to the multilayer spallation. There is a granule area with high-speed and low-density in the ejecta head. However, the degree of fragmentation is much lower in the main body of Cu. Overall, the jet masses under the four defect conditions vary greatly. Moreover, their time-dependent eject mass follow different law.

Although the Euler method we utilized to calculate the material failure under the stretch strength has met the convergence requirements for conserved quantities, the sizes of the fragmentation area and of the smallest particle are still related to the computational grid size. Our future work will be directed to large- scale computing under a micrometer and to submicron grid size. With the combination of the dynamic experimental data, the best computational size is likely to be obtained. Therefore, our simulation ability will be developed from macro to micro scale. Moreover, because of the complex thermodynamic processes during loading and unloading, establishing a correct constitutive model and EOS will be crucial to improving the reliability of out numerical simulations, and considerably more work remains to be done to study the physical factors that influence the interface instability, fragmentation, and ejecta.

The authors would like to thank the supported by “Science Challenge Project” (No. TZ2016001), the National Natural Science Foundation of China (Nos. 11372294 and 11532012), and the Foundation of National Key Laboratory of Shock Wave and Detonation Physics (No. 9140C670301150C67290).

Bai, J.S., Wang, T., Xiao, J.X., Wang, B., Chen, H., Du, L., Li, X.Z. and Wu, Q. (2017) Computational Analysis of the Metal Free-Surface Instability, Fragmentation, and Ejecta under Shock. World Journal of Mechanics, 7, 255-270. https://doi.org/10.4236/wjm.2017.79021