Pis'ma v ZhETF, vol. 102, iss. 2, pp. 89-93 © 2015 July 25
Strength of solid and molten aluminum under dynamic tension
A. E. Mayer1), P. N. Mayer
Chelyabinsk State University, 454001 Chelyabinsk, Russia
Submitted 16 March 2015 Resubmitted 17 April 2015
Under dynamic loading, an aluminum melt has a comparable or even higher strength than solid aluminum at room temperature. This phenomenon is possible due to the uniformity of the melt and the homogeneous nucleation of voids within it, whereas the fracture mechanics of a solid are controlled by pre-existing defects. High-current electron irradiation of an initially solid metal plate can be used to observe this effect.
In everyday life, solids are more capable of withstanding tension than liquids. Intuitively, this is also expected for dynamic loading of a metallic melt. Similar to the case of solid, a tensile strength of melt can be introduced as the maximal absolute value of negative pressure realized in melt under dynamic expansion before its fracture due to cavitation. It seems natural that the melt should always have less tensile strength than the corresponding solid metal. In this paper, we analyze a possible situation for aluminum in which the melt has a comparable or even higher dynamic tensile strength than the solid metal. Our analyses are based on data from experiments [1-4] and molecular-dynamics (MD) simulations [5, 6], as well as the results of our modeling. In addition, we propose a schematic for an experiment, which can be used to make measurements of the dynamic tensile strength of the molten aluminum and to observe the reported effect. This topic is not only of interest to academics, but is also of practical importance due to developing high energy density technologies.
To determine the dynamic strength of solid aluminum, we used data from high-speed impact experiments [1-3], intensive ultra-short laser irradiation , MD simulations [5, 6], and results of our MD simulations using LAMMPS  package and the potential of interatomic interactions from . In our MD simulations, the uniform uniaxial tension was modeled by the scaling of coordinates of atoms, which simulates the inertial expansion of melt typical for experimental situations (in experiments, previously compressed or heated layer firstly expands due to the pressure gradients, and then due to inertia). The system was deformed in a Nose-Hoover thermostat at the constant temperature and with the relaxation time of 0.1 ps (time step is 0.001 ps). Additional calculations in adiabatic condi-
tions showed that the tensile strength depends only on temperature at the moment of fracture. On the other hand, current temperature depends on prehistory of deformation (e.g., previous compression). Therefore, in order to have reference points, we considered the constant temperature. Periodic boundary conditions were set for all boundaries. The system was chosen large enough to ensure independence of calculated strength on the system size: the number of atoms varied from about 100 000 to 4 millions depending on the strain rate. Relaxation at zero pressure during 5ps occurred prior to tension. Temporal dependences of average longitudinal stresses in the MD system were analyzed. All data for strength of solid aluminum are collected in Fig. 1. One can see a correspondence between our MD simulations and the existing experimental  and calculated [5, 6] data for solid aluminum.
According to [4, 9], the tensile strength is a power function of the tension rate. According to [10, 11], there are two modes of fracture for solid metals: i) a heterogeneous mode, in which the damage develops from existing defects within the material (pores, microcracks), and ii) a homogeneous mode, which is realized if the concentration of the pre-existing defects is insufficient to cause failure, so that all the volume of the material takes a part in the generation of new voids from stochastic fluctuations. The transition from mode i to mode ii occurs at strain rates higher than 108 —109 s-1. This transition is characterized by a substantial drop in the strain rate sensitivity of the tensile strength. The homogeneous nucleation mode with a low strain rate sensitivity of strength corresponds to the usual behavior observed in MD simulations (see Fig. 1). In any case, the dynamic strength of solid aluminum increases from a value of approximately 2 to 6-8 GPa as the strain rate increases from ~ 106 to ~ 109 — 1010s_1, as previous experiments have shown.
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Fig. 1. Strain rate dependencies of the tensile strength of solid and molten aluminum: monocrystalline aluminum (small rings "impact 1" - impact experiments ; filled squares "MD 1" and figured rings "MD 2" - MD simulations of  and , respectively; rhombs with line "MD solid" - our MD simulations at 300 K), polycrys-talline aluminum (large rings "impact 2" and daggers "impact 3" - impact experiments [f] and , respectively; asterisks "laser" - laser experiments ), solid aluminum at 300 K with pores (markers with line "MD solid + pores" present our MD simulations, diameter of pores is 5 nm, concentrations are 5 000 and 40 000 per cubic micrometer; porosities are 0.09 and 0.7%, respectively) and homogeneous molten aluminum at 1000 K (solid line "liquid (1000 K)" - our calculations using the model , filled circles with line "MD liquid (1000 K)" - our MD simulations). Our MD simulations for solid and liquid aluminum are made with use of LAMMPS  and the interatomic potential 
For super-high strain rates (>108s the tensile strength of the melt at 1000 Iv is determined from the results of our MD simulations (see Fig. 1). The LAMMPS  package is also used with the potential of interatomic interactions from . To analyze the tensile strength of the aluminum melt in a wider range of strain rates, we used a model  based on the classical theory of a metastable liquid . The homogeneous nucleation and growth of spherical voids in the melt at negative pressures are considered in this model. The growth of the voids is described by the Rayleigh-Plesset equation  and the thermal fluctuations  provide the nucleation for super-critical voids, which are able to grow at a given negative pressure. The maximum absolute value of the negative pressure is when the growth rate of the volume of the voids is equal to the growth rate of the volume of the melt. This maximum value is the
tensile strength of the melt at the given strain rate and temperature. The calculated strain rate dependence of the tensile strength of the aluminum melt at 1000 Iv is shown in Fig. 1. In these calculations, we take into account the size distribution of the voids (all voids are divided into generations according to the time of nucleation). We used data  for surface tension, data  on viscosity of the melt, and the wide-range equation of state  to determine the thermodynamic parameters of the melt. One can see a close fit between the model  and MD in the range of strain rates > 108 s_1, where MD simulation makes sense.
Fig. 1 demonstrates our main idea. At super-high strain rates, the strength of the solid metal is higher than the strength of the melt, which is confirmed from MD simulations result and laser experiments. At strain rates less than • 107s_1, the tensile strength of the melt becomes higher than the strength of the poly-crystals. When the strain rate decreases to less than
• 106s_1, the melt becomes more durable than the monocrystalline aluminum. This phenomenon is due to a weak rate sensitivity of the strength of the melt in the homogeneous nucleation mode, whereas the solid metal transitions to the heterogeneous fracture mode and its rate sensitivity sharply increases. Therefore, a higher melt strength may be achieved if the homogeneous nucleation mode occurs in the melt over the strain rate range from 106 to 108s_1.
In order to argue our idea, Fig. 1 also presents MD simulation results of uniaxial tension with the constant strain rate of aluminum with high concentration (5 000 and 40 000 per cubic micrometer) of small pores (with diameter of 5 nm). Simulation conditions were similar to the cases of solid aluminum without defects and aluminum melt considered before. Initial pore and the void formed in the course of deformation are imaged in Fig. 2 (the program Ovito  was used for visualization). One can see a sharp drop of the tensile strength with the strain rate decrease in the case of solid metal with pores in Fig. 1. Difference in strength of perfect monocrystal and porous solid aluminum grows up with the strain rate decrease. The strength of our models of porous material becomes less than the melt strength at the strain rates < 5 • fO10 and < 1 • 1010 s_1 depending on concentration of pores. It means that softening due to the presence of defects (pores) is more substantial than the thermal softening. On the other hand, one can see in Fig. f the transition to homogeneous mode of fracture for porous aluminum with concentration of pores of 5 000 per cubic micrometer at the strain rate of 1011 s_1 - the tensile strengths of porous and perfect aluminum coincide at such strain rate. Calculations show that a decrease
© 5000 nm
40 000 nm
10 10 10 10 10 Strain rate (1/s)
Fig. 2. (Color online) Initial pore and void formed at late stage of deformation; aluminum, temperature is 300 K, strain rate is 109 s_1. Color corresponds to the value of centro-symmetry parameter; visualization with use of Ovito 
of the pore concentration leads to a shift of the considered strength drop in the direction of smaller strain rates. Therefore, the transition between homogeneous and heterogeneous modes of fracture accompanied by the strength drop substantially depe
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