English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We introduce positive elastic potentials in the harmonic approximationleading by Hamilton's variational principle to fractional Laplacian matriceshaving the forms of power law matrix functions of the simple local BornvonKarman Laplacian. The fractional Laplacian matrices are well defined onperiodic and infinite lattices in $n=1,2,3,..$ dimensions. The present approachgeneralizes the central symmetric second differenceoperator (Born von KarmanLaplacian) to its fractional central symmetric counterpart (FractionalLaplacian matrix).For non-integer powers of the Born von Karman Laplacian, thefractional Laplacian matrix is nondiagonal with nonzero matrix elementseverywhere, corresponding to nonlocal behavior: For large lattices the matrixelements far from the diagonal expose power law asymptotics leading tocontinuum limit kernels of Riesz fractional derivative type. We presentexplicit results for the fractional Laplacian matrix in 1D for finite periodicand infinite linear chains and their Riesz fractional derivative continuumlimit kernels.The approach recovers for $\alpha=2$ the well known classicalBorn von Karman linear chain (1D lattice) with local next neighborspringsleading in the well known continuum limit of classic local standardelasticity, and for other integer powers to gradient elasticity.We also presenta generalization of the fractional Laplacian matrix to n-dimensional cubicperiodic (nD tori) and infinite lattices. For the infinite nD lattice wededucea convenient integral representation.We demonstrate that our fractionallattice approach is a powerful tool to generate physically admissible nonlocallattice material models and their continuum representations.

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