Fractional Lattice Dynamics: Nonlocal constitutive behavior generated by power law matrix functions and their fractional continuum limit kernels

We introduce positive elastic potentials in the harmonic approximation leading by Hamilton's variational principle to fractional Laplacian matrices having the forms of power law matrix functions of the simple local Bornvon Karman Laplacian. The fractional Laplacian matrices are well defined on periodic and infinite lattices in $n=1,2,3,..$ dimensions. The present approach generalizes the central symmetric second differenceoperator (Born von Karman Laplacian) to its fractional central symmetric counterpart (Fractional Laplacian matrix).For non-integer powers of the Born von Karman Laplacian, the fractional Laplacian matrix is nondiagonal with nonzero matrix elements everywhere, corresponding to nonlocal behavior: For large lattices the matrix elements far from the diagonal expose power law asymptotics leading to continuum limit kernels of Riesz fractional derivative type. We present explicit results for the fractional Laplacian matrix in 1D for finite periodic and infinite linear chains and their Riesz fractional derivative continuum limit kernels.The approach recovers for $\alpha=2$ the well known classical Born von Karman linear chain (1D lattice) with local next neighbor springsleading in the well known continuum limit of classic local standard elasticity, and for other integer powers to gradient elasticity.We also present a generalization of the fractional Laplacian matrix to n-dimensional cubic periodic (nD tori) and infinite lattices. For the infinite nD lattice we deducea convenient integral representation.We demonstrate that our fractional lattice approach is a powerful tool to generate physically admissible nonlocal lattice material models and their continuum representations.
Comment: Advanced Problems in Mechanics http://www.apm-conf.spb.ru/, Jun 2016, St. Petersburg, Russia. To appear in APM 2016 proceedings http://www.apm-conf.spb.ru/, APM 2016