Finite-temperature stress calculation and phonon dispersion evolution in a uniaxially strained anharmonic crystal

R. Parthasarathy, A. Misra, L. Ouyang
Tennessee State University,
United States

Keywords: anharmonic potential, non-affine thermal vibration, cauchy-born rule, free energy, vibration tensor, vibration stress


Equivalent continua motivated by atomistic simulations are useful to study the finite temperature thermomechanical, optical, and electrical properties of nanocrystals. Such equivalence requires (a) the connection between the continuum deformation field and the lattice positions of the crystal both in terms of average positions and thermal vibration, and (b) equivalence of free energy between discrete and continuum descriptions. However, reconciling continuum stress definitions as work conjugates of deformation measures from molecular dynamics simulation is non-trivial, particularly when atomic displacements are non-affine. Even for a homogeneously deforming single crystal, we show that the evolution in thermal fluctuations of atoms is non-affine and contributes significantly to the continuum stress, particularly at high temperatures. Work conjugate stresses termed “Static Stress” and “Vibration Stress” have been derived for (a) first order deformation gradients corresponding to atomic equilibrium positions and for (b) vibration tensors corresponding to second moments of atomic position, respectively. Using MD simulation in NVT ensembles for fcc aluminum subjected to [100] uniaxial deformation, the effect of these stress measures on the mechanical behavior in the elastic range and in the vicinity of softening has been demonstrated. Particularly, the experimentally observed phenomenon of reversible vibrational softening is predicted by the derived stress measures at temperatures of 0.7Tm to 0.9 Tm. The Vibration Stress quantitatively demonstrates vibrational modes to be precursors to deformation-induced phase transition or mechanical instability, as also observed by neutron scattering experiments and ab initio simulations on crystalline solids. Under compression, the Vibration Stress goes through a softening regime prior to the onset of static non-affinity and mechanical instability. Correspondingly, the phonon group velocities also vary in a non-monotonic manner under compression. As the material approaches softening under tension, the vibration tensor demonstrates localized zones of non-affinity and anisotropy before similar localization is observed in the static atomic displacements. Anomalies in the phonon dispersion and group velocity distribution are also explained by the non-affine evolution of the thermal lattice vibrations with macroscopic deformation. The results match those from ab initio molecular dynamics simulation on single crystal Aluminum at similar temperature range. They reflect that while the deformation gradient corresponding to equilibrium atomic positions explores the global potential energy well of the system, the vibration tensor explores the local potential wells at individual atomic sites. The results show the usefulness of the derived work-conjugate pairs for multi-scale modeling of high temperature mechanical behavior of nanocrystals, since they naturally lead to a higher order continuum description. The proposed measures are applicable in general for continuum interpretation of problems involving thermally activated processes including onset of yield in crystalline solids, and phase transition of crosslinked polymers under temperature and moisture gradients.