Elastic compliances and stiffnesses of the fcc Lennard-Jones solid

Abstract

The isothermal elastic compliances, stiffnesses, and bulk moduli of a Lennard-Jones solid organized into an fcc crystal structure (256 atoms in ${4}^{3}$ unit cells) have been calculated as a function of testing temperature (expressed as the mean kinetic energy per atom). Tests conducted in pure shear were used to determine ${\mathit{S}}_{44}$ and ${\mathit{C}}_{44}$=${\mathit{G}}_{100}$, where 100 refers to crystallographic directions. Tests imposing axial elongation with fixed lateral dimensions established ${\mathit{C}}_{11}$ and ${\mathit{C}}_{12}$. Axial deformation with zero lateral pressure (a tension test) was used to determine ${\mathit{S}}_{11}$, ${\mathit{S}}_{12}$, ${\mathit{E}}_{100}$ and ${\ensuremath{\nu}}_{100}$. This provided an independent set of results for comparison with the dilatational stiffnesses ${\mathit{C}}_{11}$ and ${\mathit{C}}_{12}$. The bulk modulus K was obtained by independent triaxial tension testing. The stiffnesses, compliances, and moduli were determined by regression analysis and digital filtering applied to combinations of the stress-tensor and strain-tensor data stored at each iteration during the constant-rate deformation experiments.While the cubic fcc Lennard-Jones solid expectedly obeys the Cauchy relations for central-force potentials, it is not isotropic, allowing \ensuremath{\nu} to take on values other than 1/4 as originally proposed by Poisson. The present calculations show ${\ensuremath{\nu}}_{100}$=0.347 for the fcc Lennard-Jones solid with a Young's modulus of ${\mathit{E}}_{100}$=61.1\ensuremath{\varepsilon}/${\mathrm{\ensuremath{\sigma}}}^{3}$, an initial (as indicated by superscript 0) shear modulus of ${\mathit{G}}_{100}^{0}$=57.2\ensuremath{\varepsilon}/${\mathrm{\ensuremath{\sigma}}}^{3}$, and an initial bulk modulus of ${\mathit{K}}^{0}$=71.2\ensuremath{\varepsilon}/${\mathrm{\ensuremath{\sigma}}}^{3}$ at zero temperature. The moduli all decreased with increasing temperature. Reuss, Voigt, and Hashin and Shtrikman [J. Mech. Phys. Solids 10, 335 (1962)] bounds on the isotropic elastic properties of polycrystalline aggregates of Lennard-Jones material were also determined. Computed values of the moduli are in reasonable agreement with experimental results for solid argon and crystalline polyethylene.