Construction of kernels for nonlocal elasticity from one-dimensional dispersion data in reciprocal space

Kernels for non-local elasticity are frequently obtained from phonon dispersion relations. The dispersion relations are generally available in the form of one-dimensional (1D) curves in reciprocal space for different high-symmetry directions of the Brillouin zone, however, 3D kernels for three-dimensional (3D) solids are needed. To the best of knowledge of the authors, there is no systematic procedure to obtain material specific nonlocal kernels (for integral type nonlocality) from the dispersion data. The present paper proposes strategies to build the 3D kernels from the dispersion data. Our particular focus is on isotropic media that is interesting due to the inherent quasi-1D nature of the axisymmetric kernel. We have obtained these kernels using Fourier-Bessel transforms, yielding axisymmetric kernel profiles in both reciprocal and real spaces.