Figure 39. Mid-phase evolution from BCC to FCC.
APPLICATIONS IN IMPACT
Many computational methods work by minimizing the potential energy which yields a quasistatic model. For many cases this is sufficient, however for applications which involve a decent amount of dynamics such as thermal activation, and kinetic energy transfer, this approach is insufficient. Thermal activation can be included in stochastic modeling but not explicitly. For explicit dynamics Newton's equations of motion are usually used to model atoms or material particles. In many ways the propagation of stress waves are very important when considering damage nucleation in materials under high speed loading conditions such as an impact. To understand the material response to such conditions, analysis at the atomic scale should be conducted, to investigate the nucleation of dislocations, zones of amorphousness, nano-cracks, phase changes, etc. This atomic phenomenon can better help us to understand the high scale material behavior. There has been some work in the area of dynamic multiscale analysis with atomistically-based methods. This chapter will describe some of them and show how the GP method may be extended for use in such cases. It will also discuss how the GP method should grow to better account for certain dynamic phenomena.
A. Introduction
Continuum level modeling of dynamic impact of a thin glass sheet was simulated by Hu et al. using the peridynamic method to investigate the dynamic fracture pattern under various boundary conditions.94 Their model size was the same as the experiments at 10x10x0.3 cm3, impact velocities ranged from 61 to 200 m/s. Their results show very good qualitative agreement with the experiments. Although their work is very impressive and useful, it does not explain nor imply what atomistic phenomena is involved. Their only measure for fracture is a critical bond-stretch that correlates to the material's critical energy release rate.95
Branicio et al. used a 200 million atom MD simulation to model atomistic damage in AlN from a hypervelocity projectile impact at 15 km/s.96 They found a phase transformation, following the initial elastic compression wave, from the usual wurtzite to
a rocksalt phase which is stable at lower volume and higher energies. Behind this phase transformation wave is a source of nanocavities and kink bands. As the wave returns, being reflected from the other side of the sample, mode-I cracks nucleated from the nanocavities and mode-II from the kink band superdislocation boundaries. Although very interesting results were found there are a couple draw backs to their simulation design.
Firstly the size of their impactor had 500,000 atoms which is supposed to represent an armor pricing bullet; but was this sufficient to transfer the correct momentum and energy flux? Secondly in order to have more realistic boundary conditions their model size had to be extremely large; 200 million atom MD simulations still require a long time to run even on the most advanced supercomputers. The authors may benefit from using an atomistically-based concurrent multiscale technique that can handle the dynamics required by their impact problem.
Wang wrote a dissertation about an adaptive multiscale method for modeling nonlinear deformation in nanoscale materials based on the QC method.97 Wang proposed a remeshing technique using a critical strain or energy criterion when the homogeneity of the microstructure/ deformation was violated and implemented finite temperature into the QC method via a local harmonic approximation which integrates over all available normal modes of vibration to derive the equilibrium entropy. The example application for their method is nano-indentation using the Mixed Penalty functional in the perturbed Lagrangian to implement contact from an impactor. Even though finite temperature is implemented the simulation still maintains the quasi-static nature of the QC method, thus the atomistic domain is not governed by MD or Monte Carlo (MC) but rather Molecular Mechanics (MM). This means that high velocity impact cannot be accurately modeled since stress waves would not propagate through time, an explicit version of QC would be needed.
Lidorikis et al. proposed a concurrent multiscale method that bridges the atomistic domain with the continuum FE mesh through a linear average of each domain's Hamiltonian98,99 which is similar to the Bridging Domain Method (BDM) by Xiao and Belytschko100 as they both derive their methodologies from Broughton et al. for wave reflection suppression.101 They show that their scale coupling method is accurate for both static and dynamic cases. The dynamic case is illustrated with an impact on an Si/Si3N4
interface. Farrell and Park et al.102,103 used the Bridging scale method (BSM), although without a hand-shake region for blending the energies, to study wave and intersonic crack propagation capturing the formation of daughter cracks. They both found no wave reflection from the scale interface region and both validate their models by comparing them to the full MD simulations. These methods show an overall high accuracy for all of their capabilities; however their methods require the element size to be the same as the atomic lattice in the scale interface domain thus making it a direct coupling method (DC) which is not the most efficient for saving degrees of freedom in the continuum.
Guo et al. implemented a unique hand-shake domain composed of coarse-grain like material points; their Material Point Method (MPM) is able to smoothly transfer the atomistic information to the FE continuum via modified interpolation shape functions to reduce artificial forces on the hierarchical background grids.104 They tested the MPM by using a step-like wave and a wave packet propagating within a bar. They were able to implicitly include the short-wavelength phonons and their dissipation into the MPM region by weakly coupling the region to a Brownian heat bath whose dynamics are set to the simulation temperature by invoking equipartition and correcting for the lost degrees of freedom.101 Thus Newton's equations of motion are replaced by Langevin equations for this higher scale.
In the next section the GP method will be investigated for its ability to handle dynamic wave propagation through the use of auto-duality domains to save degree of freedom. Domains are used to decompose higher scale particles into atoms in regions that have high energies in an attempt to maintain a degree of accuracy.