Andrei A. Gusev and Sergey A. Lurie
Lagrangian variational approach is employed to derive the equations of equilibrium of continuum strain-gradient elasticity. For a periodic lamellar-morphology strain-gradient medium, we obtain an exact formula for the overall, system stiffness. We compare the formula with direct molecular-mechanics estimates of stiffness of binary Lennard-Jones crystals. Remarkably, the comparison reveals that the strain-gradient solution remains fairly accurate for all the crystals studied, including those with order of unity atoms in the crystal unit cell. Thus, one can surmise that the strain-gradient correction alone can already be sufficient to extend the validity of continuum-level elasticity to near atomistic length scales.