Temperature-dependent void nucleation and growth in single-crystal and nanocrystalline iron at extreme strain rates: atomistic insights

In this paper, we use molecular dynamics simulation with the embedded atomic method to perform triaxial deformation experiments on single-crystal and nanocrystalline iron at a strain rate of 5×10−9 s−1 and investigate the temperature effect on the void nucleation and growth process. We also evaluate the applicability of the Nucleation And Growth (NAG) model for single-crystal iron. The results indicate that the maximum tensile stress of both single-crystal and nanocrystalline iron decreases as temperature increases, with a reduction of 35.9% for single-crystal iron and 36.2% for nanocrystalline iron from 100 K to 1100 K. It is demonstrated that void nucleation and growth is more favored at high temperature. The void nucleation and growth process in single-crystal iron under high strain rate follows the NAG model. We analyze the sensitivity of the NAG parameters at different temperatures and find that the void nucleation and growth threshold of single-crystal iron is much higher than that of low carbon steel. The results can provide insights for developing fracture models of iron at extremely high strain rate and describing the dynamic damage at continuum length scales.


Introduction
Metal fracture under high-speed impact and extreme strain rates is a complex problem in fracture mechanics, drawing significant attention.This phenomenon is primarily caused by dynamic fracture mechanisms involving voids nucleation, growth, and coalescence, ultimately leading to material damage accumulation.Various factors, such as microstructure, initial defects, temperature, and strain rate, affect dynamic metal fracture.Strain rate has a substantial impact, with higher strain rates resulting in increased fracture strength.Common experimental methods, like the Split Hopkinson Pressure Bar, Gas Gun, and Laser, cover a wide range of strain rates from 10 2 to 10 10 s −1 .Numerous studies have explored dynamic metal fracture under different strain rates [1,2].For instance, Zaretsky et al. [3] examined the dynamic response of polycrystalline iron at varying initial temperatures and a strain rate of 10 5 s −1 , analyzing temperature effects.Remington et al. [4] investigated grain size impact on spalling strength at strain rates of 10 6 to 10 7 s −1 , revealing grain boundaries' role in void nucleation.
Several theoretical models, such as the Gurson model, Johnson model, and Nucleation-And-Growth (NAG) model [5], have been developed based on experimental research to describe macro-level spallation behavior.These models are primarily empirical and depend on unknown parameters due to the complexity of dynamic metal fracture, which involves various factors and operates on multiple time and length scales.
Molecular dynamics (MD) simulations have emerged as a powerful tool for understanding dynamic metal fracture at extreme strain rates.MD simulations offer microscopic insights, measuring physical quantities like temperature, tensile stress, phase transitions, and void evolution [6].K. Mayer [7] studies the maximum tensile stress of single-crystal iron at a strain rate of 5×10 9 s -1 under uniaxial tension, observing two stress peaks linked to dislocation and void nucleation.Rawat et al. [8] explored void evolution in single-crystal iron under varying strain rates, revealing rate-dependent nucleation and growth.Shao et al. [9] studied temperature effects on phase transitions in single-crystal iron at 2×10 9 s −1 .Rawat et al. [10] used MD to study temperature effects on single-crystal copper's dynamic fracture at 5×10 9 s −1 , aligning atomistic observations with the NAG model.
Despite extensive studies on dynamic fracture in metals, there is limited research on the influence of temperature at extreme strain rates and the applicability of the NAG model.This paper employs MD simulations to investigate temperature effects on void nucleation and growth in single-crystal and nanocrystalline iron at extreme strain rates and discusses the NAG model's suitability for single-crystal iron.

Computational method
We used an EAM potential provided by Mendelev [11] to model atomic interactions in iron and performed MD simulations with LAMMPS [12] to study the triaxial deformation of single-crystal and nanocrystalline iron at a strain rate of 5×10 9 s -1 .This potential is widely used in molecular dynamics simulation of iron under high strain rate [13,14].The single-crystal system had 2×10 6 atoms in a cubic domain of 28.55 nm, while the nanocrystalline system had a similar size and number of atoms, but with an average grain size of 13 nm.In order to make the study more general, nanocrystalline model was generated by Voronoi tesselation method with grains at random positions and orientations.Periodic boundary conditions were applied in all directions.The time-step was 1 fs.We relaxed the systems in the NPT ensemble for 30 ps at temperatures ranging from 100 K to 1100 K, and then triaxially expanded them at a constant strain rate.Figure 1(a) and (b) show the equilibrated systems of single-crystal and nanocrystalline iron, respectively.We used Ovito for visualization.
It should be noted that the interatomic potentiall [11] we used in this study may not correctly describe α-ε phase transition under the shock at about 13 GPa, which is a common issue for many empirical potentials for iron [15].Our main focus in this research is on the void nucleation and growth mechanisms in single-crystal and nanocrystalline iron at extreme strain rates, which are independent of the phase transition.

Temperature effect on tensile stress
We calculate the tensile stress of the simulation system.The stress-time profile reflects an overall response of single-crystal and nanocrystalline iron, as can be seen in Figure 2 (a) and (b), respectively.Figure 2 shows that the tensile stress of single-crystal iron is higher than nanocrystalline iron.Molecular dynamics simulation shows that the maximum tensile stress of single-crystal iron is about 18.3 GPa and that in nanocrystalline iron is 14.58 GPa at 300 K. Mayer [7] studied the maximum tensile stress of single-crystal iron at a strain rate of 5×109 s-1 under uniaxial tension.The maximum tensile stress measured in his research is 17.6 GPa.Ashitkov et al. [16] measured the maximum tensile stress in the spallation test of iron (sample thickness 50mm) from 17.8-20.3GPa.The maximum tensile stress-temperature profile in single-crystal and nanocrystalline iron at 100-1100 K is shown in Figure 3. From Figure 3 we can find that as the temperature increases, the maximum tensile stress of single-crystal and nanocrystalline iron decreases gradually.For single-crystal iron, the maximum tensile stress is 21.4 GPa at 100 K, and the minimum tensile stress is 13.7 GPa at 1100 K, with a reduction of 35.9%.For nanocrystalline iron, the maximum tensile stress is 16.82 GPa at 100 K, and the minimum tensile stress is 10.75 GPa at 1100 K, with a reduction of 36%.The result indicates that the temperature effect on the maximum tensile stress of single-crystal and nanocrystalline iron is significant.On the other hand, the change of strain rate has no noticeable effect on the tensile stress peak of single-crystal iron [8].

Temperature effect on the void volume fraction
As observed in Figure 4 (a) and (b), the variation of the overall void volume fraction in single-crystal and nanocrystalline iron is similar.For single-crystal iron, with the increase of temperature, the time of void nucleation is advanced, and the rate of increase of void volume fraction is accelerated.The overall void volume fraction at 100-1100 K is almost the same at 30ps, which is about 29%.For nanocrystalline iron, however, the overall void volume fraction is about 33% at 30ps, which is more significant than that in single-crystal iron.Figure 5 shows the void nucleation time in nanocrystalline and single-crystal iron in 100-1100 K.As observed from Figure 5, the void nucleation time in nanocrystalline iron is earlier than that in singlecrystal iron, it is about 7 ps earlier than single-crystal iron at 100 K, and 6.5 ps at 1100 K.This indicates that the temperature effect is significant, and the void nucleation time, and the void nucleation time advances with the increase of temperature.We can also find that based on room temperature, the effect of temperature reduction on void nucleation time is more significant than the temperature increase in nanocrystalline and single-crystal iron.For single-crystal iron, when the temperature rises from 300 K to 1100 K, and the void nucleation time differs by 2.5 ps, the void nucleation time varies by 2.1 ps while the temperature drops to 100 K.For nanocrystalline iron, when the temperature rises from 300 K to 1100 K, the void nucleation time differs by 0.8 ps, when the temperature drops to 100 K, the void nucleation time varies by 3 ps.100-1100 K). Figure 6 shows the early stages of voids nucleation, growth, and rapid coalescence in single-crystal and nanocrystalline iron at 100 K and 1100 K at 5×109 s−1 strain rate, respectively.For each system, there are two visualization schemes shown.The top row is colored according to the potential energy with the help of common neighbor analysis (CNA), we slice in the middle of the system and the slice width is 10 Å.The bottom row is colored by surface meshing tools in OVITO to visualize voids in the whole system.As observed from Figure 6, we can find that the total volume of the voids in the system is the accumulation of nucleation, growth, and coalescence of multiple independent voids.Voids nucleate randomly in single-crystal iron; however, voids nucleate along the grain boundaries in the nanocrystal iron.
In single crystal, the critical vacancy concentration for void formation depends on the temperature and strain rate.The vacancies tend to cluster and coalesce into larger voids, which then grow and link up to form cracks.In contrast, in polycrystalline samples, the role of vacancies is negligible, as the grain boundaries act as preferential sites for void nucleation and growth.

Temperature effect on nucleation and growth (NAG) model parameters
The NAG model is a microscopic physical model that is mainly used to describe the damage process occurring in the nucleation and growth of voids in metallic materials.
In the NAG model, when P s (the tensile stress) exceeds P n0 (void nucleation threshold stress), void begins nucleate.The void nucleation rate is given by Eq. ( 1) and ( 2). ( (2) In Eq. ( 1) and (2), N 0 is the threshold nucleation rate; P 1 is the stress sensitivity for void nucleation.Both N 0 and P 1 are material constants.Note that P s is the tensile stress and not the average stress in the system.
During time interval Δt, the volume of the voids nucleated is defined as Eq. ( 3).
(3) R n is the nucleation size parameter.R n is a material constant, too.

   
When the tensile stress exceeds P g0 (the threshold for void growth), the nucleation voids grow.The new void volume considering void growth is defined as Eq. ( 4).
(4) V v0 is the void volume at the beginning of the time interval Δt. η is the material viscosity.The total void volume due to nucleation and growth of voids at the end of the time interval Δt is defined as Eq. ( 5).
(5) The relative error φ is defined as Eq. ( 6). ( 6) V MD is the total void volume obtained by MD simulation.V NAG is the total void volume obtained by the NAG model calculate.
The square root of average squared relative error Σ is defined as (7) Figure 7.Comparison of void volume fraction between the NAG model and MD simulation at 100-1100 K.For any given set of NAG parameters, the total void volume as a function of time can be obtained on the premise of the known tensile stress time history curve.The parameter optimal fitting was performed by machine learning.The NAG model parameters obtained by optimal fitting were used to calculate the void volume fraction.The volume fraction changes and is compared with the results of the MD simulation, as shown in Figure 7.It can be seen from Figure 7 that the evolution of the void volume fraction of single-crystal iron at different temperatures is in good agreement with the NAG model.
Table 1 shows the parameters from MD simulation in this research and experiment [16][17].The void nucleation and growth behavior of single-crystal iron differs significantly from that of low carbon steel.Single-crystal iron has a much higher void nucleation threshold (P n0 ), a lower sensitivity coefficient (P 1 ) of the nucleation tensile stress, and a much lower material viscosity (η) than low carbon steel.These differences are attributed to the absence of grain boundaries and other defects in single-crystal iron, which hinder the nucleation and growth of voids.The temperature effect is similar for both materials, as higher temperatures increase the atomic mobility and reduce the void nucleation and growth thresholds (P n0 and P g0 ).The ratio of nucleation threshold to growth threshold is comparable for both materials, being about 5-6 times.[17].
It should be noticed that the void nucleation threshold of single-crystal iron is much higher than that of low-carbon steel, with a difference of more than 10 times.This might because low-carbon steel is a polycrystalline structure with a large number of defects, while these defects do not exist in single-crystal iron, voids are more easily nucleated at the grain boundaries [18].

Conclusion
Using molecular dynamics simulations, we explored the dynamic response of single-crystal and nanocrystalline iron subjected to triaxial tensile loading at high strain rates and temperatures.We found that the maximum tensile stress decreased with increasing temperature, while the void volume fraction increased in three stages.We also employed the NAG model to single-crystal iron and achieved a good agreement with the simulation results.We compared the parameters of the NAG model for single-crystal iron and low carbon steel, and analyzed the effects of temperature, grain boundaries, and defects on the void nucleation and growth behavior.Nevertheless, there remains a large difference between some NAG parameters derived from experimental data and MD simulations, which requires further study.

Figure 3 .
Figure 3. Relationship of maximum pressure and temperature in single-crystal andnanocrystalline.The maximum tensile stress-temperature profile in single-crystal and nanocrystalline iron at 100-1100 K is shown in Figure3.From Figure3we can find that as the temperature increases, the maximum tensile stress of single-crystal and nanocrystalline iron decreases gradually.For single-crystal iron, the maximum tensile stress is 21.4 GPa at 100 K, and the minimum tensile stress is 13.7 GPa at 1100 K, with a reduction of 35.9%.For nanocrystalline iron, the maximum tensile stress is 16.82 GPa at 100 K, and the minimum tensile stress is 10.75 GPa at 1100 K, with a reduction of 36%.The result indicates that the temperature effect on the maximum tensile stress of single-crystal and nanocrystalline iron is significant.On the other hand, the change of strain rate has no noticeable effect on the tensile stress peak of single-crystal iron[8].

Figure 4 .
Figure 4. Void volume fraction profiles for: (a) single-crystal iron; (b) nanocrystalline iron.Figure5shows the void nucleation time in nanocrystalline and single-crystal iron in 100-1100 K.As observed from Figure5, the void nucleation time in nanocrystalline iron is earlier than that in singlecrystal iron, it is about 7 ps earlier than single-crystal iron at 100 K, and 6.5 ps at 1100 K.This indicates that the temperature effect is significant, and the void nucleation time, and the void nucleation time advances with the increase of temperature.We can also find that based on room temperature, the effect of temperature reduction on void nucleation time is more significant than the temperature increase in nanocrystalline and single-crystal iron.For single-crystal iron, when the temperature rises from 300 K to 1100 K, and the void nucleation time differs by 2.5 ps, the void nucleation time varies by 2.1 ps while the temperature drops to 100 K.For nanocrystalline iron, when the temperature rises from 300 K to 1100 K, the void nucleation time differs by 0.8 ps, when the temperature drops to 100 K, the void nucleation time varies by 3 ps.

Figure 5 .
Figure 5.Time of void nucleation for: (a) single-crystal iron; (b) nanocrystalline iron (5×10 9 s −1 ,100-1100 K).Figure6shows the early stages of voids nucleation, growth, and rapid coalescence in single-crystal and nanocrystalline iron at 100 K and 1100 K at 5×109 s−1 strain rate, respectively.For each system, there are two visualization schemes shown.The top row is colored according to the potential energy with the help of common neighbor analysis (CNA), we slice in the middle of the system and the slice width is 10 Å.The bottom row is colored by surface meshing tools in OVITO to visualize voids in the whole system.As observed from Figure6, we can find that the total volume of the voids in the system is the accumulation of nucleation, growth, and coalescence of multiple independent voids.Voids nucleate randomly in single-crystal iron; however, voids nucleate along the grain boundaries in the nanocrystal iron.In single crystal, the critical vacancy concentration for void formation depends on the temperature and strain rate.The vacancies tend to cluster and coalesce into larger voids, which then grow and link up to form cracks.In contrast, in polycrystalline samples, the role of vacancies is negligible, as the grain boundaries act as preferential sites for void nucleation and growth.

Figure 6 .
(a) single-crystal 100 K (b) single-crystal 1100 K (c) nanocrystalline 100 K 3 (d) nanocrystalline 1100 K Evolution of voids nucleation, growth in single-crystal and nanocrystalline iron at 100 K and 1100 K.

Table 1 .
Best-fit NAG parameters in single-crystal iron at 100-1100 K.