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In this paper, the self-adjointness of Eringen's nonlocal elasticity is investigated based on simple one-dimensional beam models. It is shown that Eringen's model may be nonself-adjoint and that it can result in an unexpected stiffening effect for a cantilever's fundamental vibration frequency with respect to increasing Eringen's small length scale coefficient. This is clearly inconsistent with the softening results of all other boundary conditions as well as the higher vibration modes of a cantilever beam. By using a (discrete) microstructured beam model, we demonstrate that the vibration frequencies obtained decrease with respect to an increase in the small length scale parameter. Furthermore, the microstructured beam model is consistently approximated by Eringen's nonlocal model for an equivalent set of beam equations in conjunction with variationally based boundary conditions (conservative elastic model). An equivalence principle is shown between the Hamiltonian of the microstructured system and the one of the nonlocal continuous beam system. We then offer a remedy for the special case of the cantilever beam by tweaking the boundary condition for the bending moment of a free end based on the microstructured model.

We analyze time-discrete and continuous 'fractional' random walks on undirected regular networks with special focus on cubic periodic lattices in n = 1, 2, 3, .. dimensions. The fractional random walk dynamics is governed by a master equation involving fractional powers of Laplacian matrices L α 2 where α = 2 recovers the normal walk. First we demonstrate that the interval 0 < α ≤ 2 is admissible for the fractional random walk. We derive analytical expressions for fractional transition matrix and closely related the average return probabilities. We further obtain the fundamental matrix Z (α) , and the mean relaxation time (Kemeny constant) for the fractional random walk. The representation for the fundamental matrix Z (α) relates fractional random walks with normal random walks. We show that the fractional transition matrix elements exihibit for large cubic n-dimensional lattices a power law decay of an n-dimensional infinite space Riesz fractional derivative type indicating emergence of Lévy flights. As a further footprint of Lévy flights in the n-dimensional space, the fractional transition matrix and fractional return probabilities are dominated for large times t by slowly relaxing long-wave modes leading to a characteristic t − n α -decay. It can be concluded that, due to long range moves of fractional random walk, a small world property is emerging increasing the efficiency to explore the lattice when instead of a normal random walk a fractional random walk is chosen.

We analyze a random walk strategy on undirected regular networks involving power matrix functions of the type L α 2 where L indicates a 'simple' Laplacian matrix. We refer such walks to as 'Fractional Random Walks' with admissible interval 0 < α ≤ 2. We deduce for the Fractional Random Walk probability generating functions (network Green's functions). From these analytical results we establish a generalization of Polya's recurrence theorem for Fractional Random Walks on d-dimensional infinite lattices: The Fractional Random Walk is transient for dimensions d > α (recurrent for d ≤ α) of the lattice. As a consequence for 0 < α < 1 the Fractional Random Walk is transient for all lattice dimensions d = 1, 2, .. and in the range 1 ≤ α < 2 for dimensions d ≥ 2. Finally, for α = 2 Polya's classical recurrence theorem is recovered, namely the walk is transient only for lattice dimensions d ≥ 3. The generalization of Polya's recurrence theorem remains valid for the class of random walks with Lévy flight asymptotics for long-range steps. We also analyze for the Fractional Random Walk mean first passage probabilities, mean residence times, mean first passage times, and global mean first passage times (Kemeny constant). For the infinite 1D lattice (infinite ring) we obtain for the transient regime 0 < α < 1 closed form expressions for the fractional lattice Green's function matrix containing the escape and ever passage probabilities. The ever passage probabilities fulfill Riesz potential power law decay asymptotic behavior for nodes far from the departure node. The non-locality of the Fractional Random Walk is generated by the non-diagonality of the fractional Laplacian matrix with Lévy type heavy tailed inverse power law decay for the probability of long-range moves. This non-local and asymptotic behavior of the Fractional Random Walk introduces small world properties with emergence of Lévy flights on large (infinite) lattices.