This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (

Volumetric strain can be divided into two parts: strain due to bond distance change and strain due to vacancy sources and sinks. In this paper, efforts are focused on studying the atomic lattice strain due to a vacancy in an FCC metal lattice with molecular dynamics simulation (MDS). The result has been compared with that from a continuum mechanics method. It is shown that using a continuum mechanics approach yields constitutive results similar to the ones obtained based purely on molecular dynamics considerations.

In a work published in 1976, Blech [

The change in volume describing a vacancy is related to the original atomic volume by the parameter _{v}

Our work attempts to provide a more accurate representation of this relaxation factor, both in atomic scale and in continuum mechanics scale. By comparing results from methods applicable in different length domains, constitutive values are sought for multiscale material modeling.

The interactions between aluminum atoms in our simulation are characterized by Daw and Baskes' embedded-atom method [

Daw, Foiles, and Baskes [

Here, _{i}_{h,i}_{h}_{,}_{i}

These atomic densities are, as shown in

The embedding function, _{i}_{ij}

The virial definition of atomic stress is used to calculate the stress around a given volume of simulation space:

Here, F_{ij} is the force on an atom

There have been many researchers who have questioned point-wise stress calculation in a system using the expression for atomic stress taken from the virial theorem. Zimmerman

For the aluminum simulations, the LAMMPS molecular dynamics simulation software package was used [

The system was first allowed to converge to equilibrium, which we simulated for 40 picoseconds (ps). Following achieving a convergence, atomic positions and point-wise stresses were collected every 0.5ps for duration of 10.0 ps under NVT conditions. Next, a void was created by removing an atom from the lattice. Data collection continued for 10.0 ps. Atom specific positions and stresses were collected for the original atom's 12 first nearest-neighbors before and after atom removal.

The volumetric strain created after the removal of an atom is found by direct measurement of first-nearest neighbor positions. Prior to the vacancy formation, distances between each first-nearest neighbor and the atom to be removed were recorded every 0.5ps (500 time steps), for 10.0ps (10^{3} time steps). Averaging the first-nearest neighbor positions, we can find the center of the void, and from there an average neighbor distance from the void, R_{1}, is found. Next, the atom is removed and the system is allowed to converge to an equilibrium which took about 500 time steps. Similar to before, distances between each first-nearest neighbor and the center of the void are recorded every 500 time steps, for 10^{3} time steps. A second average neighbor distance, R_{2}, is found. Spherical volumes based on these two radii are computed and the volumetric strain is computed as shown below.

The average initial (R1) and final (R2) distances to the atom or void center for one particular molecular dynamics run are shown in _{I} = 0.060 +/–0.013. The error found here is based on uncertainty in (R1) and (R2), propagated through

In this section, continuum mechanics formulations are introduced to calculate the spherical stress induced by the removal of a matrix atom. The location of the missing atom is simplified as a spherical cavity inside an infinite elastic body. The interactions between atoms, including short range repulsion and long distance attraction, are the source of the stresses in continuum level. After the sudden removal of an atom, the attraction can no longer be balanced by the repulsion. Hence the atoms nearby will sink into the void until they reach another balance. This is the mechanism of shrinkage strain at the missing atom site. In this method, the atoms interactions are treated as hydrostatic pressure around the cavity. By introducing the elastic constitutive relationship, the volumetric stress can be calculated, which is shown in

Shown in

By Hooke's Law in spherical coordinates, we have:

Applying _{v}/C_{a}≈1×10^{–5}

According to Blech’s relationship [

The vacancy generation/annihilation has the following rate dependent form:
_{s}

From _{x}=σ_{y}=–_{xy}=0_{z}_{x}=ε_{y} =–^{–4}, spherical stress ^{spherical} =–_{V} =–^{–4}.

By taking the vacancies annihilation induced strain into consideration, the stress/strain state is calculated to be _{z}=_{x}=ε_{y} =–^{–4}, ^{spherical}=_{V}=^{–4}. The contraction due to vacancies annihilation reduces the compressive stresses in Z direction (normal to the plane). As a plane strain element, it is assumed strain in the direction normal to the plane is zero. In other words, there are constraints to prevent the deformation in Z direction. When the density of lattice sites decrease under compressive stress, the element tends to contract in all directions and thus reduces the reaction forces in Z direction.

By changing the sign of the pressure applied, it is found that the model with vacancies generation/annihilation mechanism yields more tensile strain than the one without does. Therefore, it can be concluded that the bulk modulus κ is composed by two parts:
_{e} is due to the interaction between atoms. Tensile strain increases the bond distances between atoms and yields an overall tensile stress; compressive strain shortens the bond distances and results in compressive stress.

κ_{v} reflects the change due to the vacancy concentration. Tensile stress exaggerates the grain boundaries by producing more vacancies; compressive stress tends to merge the grain boundaries and thus reduces vacancies. Both generation and annihilation of vacancies result in the corresponding volumetric strain. κ_{v} can be derived from

In case of loading time is much larger than vacancy relaxation time, as usually is, κ_{v} is much smaller than κ_{e}. In this example, κ=59 GPa, κ_{e}=60.7 GPa, and κ_{v}=–1.75 GPa. κ_{v} is only 2.7% of κ_{e,} which makes it reasonable to approximate κ_{e}

Using LAMMPS molecular dynamics simulator with the embedded-atom method, we simulated an aluminum lattice at 533K. We outputted atom positions and virial stresses for a particular atom and its first-nearest neighbors. That particular atom was then removed, and we used the change in positions to calculate the volume strain due to the creation of a void. We also calculated the volumetric strain induced spherical stress with continuum mechanics constitutive. The comparison of mechanical stress and virial stress shows that reduction factor is needed in order to bridge material modeling methods applicable in atomic scale to macroscale.

We also report that bulk modulus can be divided into two parts: one due to atomic bonds length and the other induced by vacancy sinks and sources mechanism. The latter part is strain rate dependent which can be negligible at static load.

This material is based upon work partially supported by the National Science Foundation under Grant No. CMMI-0508854 and Office of Naval Research Advanced Electrical Power Systems Program Grant N000140410778. Help received from Terry Ericsen of ONR greatly appreciated.

A plot of first-nearest neighbor distance from center of an atom (or void), versus simulation time steps in molecular dynamic simulations. Filled black circles indicated a full lattice and open circles indicate a vacancy, where the atom is removed at 10 ps into the data collection run. Average neighbor positions before and after atom removal are 2.891 +/–0.009 and 2.831 +/–0.010, respectively.

Void model in continuum mechanics domain.

Free body diagram under spherical coordinate system.

Plane strain element under compressive load.

Al material properties used in simulations.

Young's modulus, (111) texture | 6.6x10^{4} MPa at (533K) | |

Poisson’s Ratio | 0.3496 | |

Volume per Al atom, bulk | 1.38x10^{–23} cm^{3} |

Vacancy relaxation factors as reported by authors.

0.60 | |

0.20 | |

0.17 | |

0.10 | |

0.060 |