View source for Void growth and coalescence in single crystal nickel

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<h3>Abstract</h3>
Molecular dynamics simulations using Modified Embedded Atom Method (MEAM) potentials were performed to analyze material length scale influences on damage progression of single crystal nickel. Damage evolution by void growth and coalescence was simulated at very high strain rates (108–1010/s) involving four specimen sizes ranging from _5000 to 170,000 atoms with the same initial void volume fraction. 3D rectangular specimens with uniform thickness were provided with one and two embedded cylindrical voids and were subjected to remote uniaxial tension at a constant strain rate. Void volume fraction evolution and the corresponding stress–strain responses were monitored as the voids grew under the increasing applied tractions.
The results showed that the specimen length scale changes the dislocation pattern, the evolving void aspect ratio, and the stress–strain response. At small strain levels (0–20%), a damage evolution size scale effect can be observed from the damage-strain and stress–strain curves, which is consistent with dislocation nucleation argument of Horstemeyer et al.<ref>Horstemeyer, M.F., Baskes, M.I., Plimpton, S.J., 2001a. Length scale and time scale effects on the plastic flow of FCC metals. Acta Mater. 49, pp. 4363–4374</ref> playing a dominant role. However, when the void volume fraction evolution is plotted versus the applied true strain at large plastic strains (>20%), minimal size scale differences were observed, even with very different dislocation patterns occurring in the specimen. At this larger strain level, the size scale differences cease to be relevant, because the effects of dislocation nucleation were overcome by dislocation interaction.
This study provides fodder for bridging material length scales from the nanoscale to the larger scales by examining plasticity and damage quantities from a continuum perspective that were generated from atomistic results. 

Author(s): G.P. Potirniche, [http://www.hpc.msstate.edu/directory/information.php?eid=63 Mark F. Horstemeyer], G.J. Wagner, and P.M. Gullett

Corresponding Author: [mailto:mfhorst@me.msstate.edu Mark F. Horstmeyer]
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Initial geometry and BCs
[[File:yp_initial geom.png|400px|thumb|center|Fig. 1. Geometry of void growth specimen (with single void at the center) subjected to uniaxial tension under constant strain rate.]]
[[File:yp_initial geom01.png|400px|thumb|center|Fig. 2. Geometry of void growth specimen (with two voids) subjected to uniaxial tension under constant strain rate.]]
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<h3>Introduction</h3>
At macroscale, engineering materials fail by nucleation, growth and coalescence of voids. Accumulation of such voids, cavities, or cracks inside of the material is called a <i>damage</i>, or porosity or void volume fraction. The formula that defines damage is given by: &Phi; = V<sub>v</sub>/V, where V<sub>v</sub> is the volume of voids, and V is the volume of the aggregate material. New voids are introduced in the material due to increased effective stress. The increment of void volume fraction is due to void nucleation, void growth and void coalescence. This study is focused on void growth and void coalescence. The study on single void growth and multiple voids coalescence at the nanoscale is performed using molecular dynamics simulations with MEAM potentials. [[Molecular Dynamics|Molecular dynamics]](MD) is a powerful computational method to simulate elastoplastic deformation and material failure at the nanoscale.
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<h3>Material Model</h3>

The study for  was performed for four specimens of increasing size in FCC single crystal nickel. <i><u>Figure 1</u></i> shows the geometry and boundary conditions for the study of single void growth in nickel single crystal. The specimen is a plate with a central hole subjected to uniaxial strain rate. In MD, the specimens contained a number of atoms ranging from N = 4408 atoms to N = 171,376 atoms. Similarly, <i><u>Figure 2</u></i> shows the geometry and boundary conditions for the study of voids coalescence. The specimens contained two voids with initial void volume fraction equal to that of one-void specimens. The number of atoms in the specimens ranges from N = 5052 atoms to N = 175,172 atoms. The geometrical dimensions for both one-void and two-void specimens ranged from a few nanometers to tens of nanometers.
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<h3>Results and Discussion</h3>
The void growth phenomena were captured in the study using molecular dynamics principles. For the purpose of studying void growth mechanisms, a strain rate of 10<sup>10</sup>/s was considered. The specimens were loaded with uniaxial tension until total applied strain reached 41% true strain. The failure of material was seen in both single void and two void specimens. <i><u>Figure 3</u></i> shows the stress-strain curve of the one-void nickel specimen with N = 4,408 atoms indicating the stages of the specimens' internal evolution of damage. From Fig. 3, it can be observed that uniaxial stress-strain response for one-void specimen at the lowest material length comprises an elastic portion up to a true strain 14.5% and a value of the yield strength of about 20.5 GPa. Void nucleation and void growth lead to the increase of void volume fraction. The stress resistance of specimen decreased significantly to about 5 GPa at the final true strain of 41%. The necking starts at about 25% strain level. <i><u>Figure 4</u></i> presents the stress-strain response for the one-void specimen and the next largest specimen size (N = 18,448 atoms). Fig. 4< shows that the dislocation pattern for the larger specimen  is different than the dislocation pattern for smallest material length scale. The point of dislocation nucleation varies as the material length scale is changed. Another observed feature is the change in aspect ratio of the changing void. 
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<i><u>Figure 5</u></i> shows the dislocation motion as a function of the stress-strain response indicating a yield stress of 18.92 GPa (at 14.9% strain) for the lowest material length scale (N=5052 atoms) for two-void specimen. The dislocation nucleates in the ligament region between two voids. The ligament region shrank with the increase in applied strain and the voids coalesced. Afterwards, the dislocation moved towards the edges of the specimen causing the initial necking. <i><u>Figure 6</u></i> presents the stress-strain curve for next increasing length scale with N = 19,784 atoms. The yield strength decreased to 15.37 GPa with a corresponding total true strain of 12.11%. 
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<i><u>Figure 7-8</u></i> presents the evolution of void volume fraction during uniaxial straining (10<sup>10</sup>/s strain rate) for one-void specimens and two-void specimens respectively. For both one-void specimens and two-void specimens, the smaller length scale specimen experienced an increased void-volume fraction resulting from larger elastic stresses. This void volume size scale effect could be influenced by the differences in the dislocation nucleation pattern. It is seen that when the ratio of elastic to plastic strains is high, a length scale effect is observed, but when the elastic to plastic ratio is low, no length scale effect is observed. 
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<i><u>Figure 9</u></i> shows a comparison of stress-strain responses of one-void  specimens and two-void specimens for different specimen sizes. Fig. 9(a) demonstrates that the initial yield decreases monotonically from 20.5 to 14.5 GPa as the specimen size increases from 4408 to 171,376 atoms. After the initial yielding, the stress-strain response exhibited a sharp stress drop-off. Fig. 9(b) indicates similar response for two-void case with a yield decrease from 18.92 to 12.2 GPa, as the specimen size increases from 5052 to 175,172 atoms. <i><u>Figure 10</u></i> illustrates that the initial yield strength for the two-void specimens is consistently lower than the initial yield strength for the one-void specimens. Also from Fig. 10, the lowering of stress-strain curve for the two-void specimens than for the one-void specimens was observed. 

Triaxility variations with uniaxial applied strain, at the smallest and the largest specimen sizes are presented in <i><u>Fig. 12(a) and (b)</u></i> for the one-void and two-void specimens, respectively. The fluctuation in triaxiality values around 0.8 was observed for both cases. Similarly, <i><u>Figure 11(a)</u></i> shows that for the one-void specimens, the yield stress increased from 15 to 20.5 GPa as the strain rate was increased two orders of magnitude. The same qualitative conclusion apply to the analysis of the stress-strain responses for two-void specimens as seen in Fig. 12(b).
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[[File:yp_result.png|200px|thumb|left|Fig. 3. Average stress versus true strain curve at 10<sup>10</sup>/s strain rate for the one-void nickel specimen with N = 4,408 atoms.]]

[[File:yp_result01.png|200px|thumb|right|Fig. 4. Average stress versus true strain curve at 10<sup>10</sup>/s strain rate for the one-void nickel specimen with N = 18,448 atoms.]]

[[File:yp_result02.png|200px|thumb|left|Fig. 5. Average stress versus true strain curve at 10<sup>10</sup>/s strain rate for the one-void nickel specimen with N = 5,052 atoms.]]

[[File:yp_result03.png|200px|thumb|right|Fig. 6. Average stress versus true strain curve at 10<sup>10</sup>/s strain rate for the one-void nickel specimen with N = 19,784 atoms.]]

[[File:yp_result04.png|200px|thumb|left|Fig. 7. Evolution of void volume fraction during uniaxial straining (10<sup>10</sup>/s strain rate) for one-void specimen.]]

[[File:yp_result05.png|200px|thumb|right|Fig. 8. Evolution of void volume fraction during uniaxial straining (10<sup>10</sup>/s strain rate) for two-void specimens.]]

[[File:yp_result06.png|200px|thumb|left|Fig. 9. Comparison of average stress versus true strain response (10<sup>10</sup>/s strain rate) at increasing material length scale: (a) one-void specimen; (b) two-void specimens.]]

[[File:yp_result07.png|200px|thumb|right|Fig. 10. Comparison of average stress versus true strain response (10<sup>10</sup>/s strain rate) due to void growth and void coalescence mechanisms at increasing material length scales: (a) ~5000 atoms; (b) ~19,000 atoms; (c) ~75,000 atoms; (d) ~175,000 atoms.]]

[[File:yp_result08.png|200px|thumb|left|Fig. 11. Triaxiality variation during uniaxial straining at 10<sup>10</sup>/s strain rate for: (a) one-void specimens; (b) two-void specimens.]]

[[File:yp_result09.png|200px|thumb|right|Fig. 12. Influence of strain-rate on the average stress versus true strain response for: (a) one-void specimen (N =  4880 atoms); (b) two-void specimen (N = 5052 atoms).]]

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<h3>Conclusion</h3>
Molecular dynamics simulations showing damage evolution at four increasing specimen sizes were performed to study the evolution of void growth and coalescence under very high strain rates. One-void specimens and two-void specimens were increased in size to represent an increasing length scale. Given that only uniaxial tension of one crystalline orientation free of initial defects was used, the main conclusions are the following:
* Not much nanoscale coalescence was observed, due probably to the high strain rate effects. Void growth was much more dominant than void coalescence.
* At the nanoscale, the total void growth is size-scale dependent in the elastic regime but size-independent in the plastic regime.
* The averaged axial stress-strain response clearly indicate a length scale effect regardless of the number of voids from 0% to 20% strain.
* At the nanoscale, the decrease in the stress-strain curve is due mainly to the dislocation nucleation, but no major differences are noticed in the void volume fraction evolution.
* Even though the initial void volume fraction was the same for all specimens, the differences in plastic slip nucleation pattern indicated a varying void shape evolution as the material length scale was increased. For the one-void specimens, the study revealed that the void embedded in a single crystal grew due to the crystallographic plastic slip that initiated at the hole as a consequence of the stress concentration and propagated toward the outer edges of the specimen. The nucleation and evolution pattern of the plastic slip is highly dependent on the specimen size and hence length scale of the specimen. For the two-void specimens, the study revealed that the configuration of plastic slip also influenced the void coalescence patterns but not the total void volume fractions, depending on the specimen size (length scale).
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<h3>Acknowledgments</h3>
The authors are grateful to the Center for Advanced Vehicular Systems at Mississippi State University for supporting this study. The work of G.J. Wagner and P.M. Gullett is supported by U.S. DOE Contract AC04-94AL85000.
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<h3>References</h3>

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[[Category: Research Paper]]
[[Category: nanoscale]]
[[Category: Nickel]]
[[Category: Molecular Dynamics]]