Long-range forces in controlled systems

This thesis investigates new phenomena due to long-range forces and their effects on different multi-DOFs systems. In particular the systems considered are metamaterials, i.e. materials with long-range connections. The long-range connections characterizing metamaterials are part of the more general framework of non-local elasticity. In the theory of non-local elasticity, the connections between non-adjacent particles can assume different configurations, namely one-to-all, all-to-all, all-to-all-limited, random-sparse and all-to-all-twin. In this study three aspects of the long-range interactions are investigated, and two models of non-local elasticity are considered: all-to-all and random-sparse. The first topic considers an all-to-all connections topology and formalizes the mathematical models to study wave propagation in long-range 1D metamaterials. Closed forms of the dispersion equation are disclosed, and a propagation map synthesizes the properties of these materials which unveil wave-stopping, negative group velocity, instability and non-local effects. This investigation defines how long-range interactions in elastic metamaterials can produce a variety of new effects in wave propagation. The second one considers an all-to-all connections topology and aims to define an optimal design of the long-range actions in terms of spatial and intensity distribution to obtain a passive control of the propagation behavior which may produces exotic effects. A phenomenon of frequency filtering in a confined region of a 1D metamaterial is obtained and the optimization process guarantees this is the best obtainable result for a specific set of control parameters. The third one considers a random-sparse connections topology and provides a new definition of long-range force, based on the concept of small-world network. The small-world model, born in the field of social networks, is suitably applied to a regular lattice by the introduction of additional, randomly selected, elastic connections between different points. These connections modify the waves propagation within the structure and the system exhibits a much higher propagation speed and synchronization. This result is one of the remarkable characteristics of the defined long-range connections topology that can be applied to metamaterials as well as other multi-DOFs systems. Qualitative experimental results are presented, and a preliminary set-up is illustrated. To summarize, this thesis highlights non-local elastic structures which display unusual propagation behaviors; moreover, it proposes a control approach that produces a frequency filtering material and shows the fast propagation of energy within a random-sparse connected material

Helical states of nonlocally interacting molecules and their linear stability: geometric approach

The equations for strands of rigid charge configurations interacting nonlocally are formulated on the special Euclidean group, SE(3), which naturally generates helical conformations. Helical stationary shapes are found by minimizing the energy for rigid charge configurations positioned along an infinitely long molecule with charges that are off-axis. The classical energy landscape for such a molecule is complex with a large number of energy minima, even when limited to helical shapes. The question of linear stability and selection of stationary shapes is studied using an SE(3) method that naturally accounts for the helical geometry. We investigate the linear stability of a general helical polymer that possesses torque-inducing non-local self-interactions and find the exact dispersion relation for the stability of the helical shapes with an arbitrary interaction potential. We explicitly determine the linearization operators and compute the numerical stability for the particular example of a linear polymer comprising a flexible rod with a repeated configuration of two equal and opposite off-axis charges, thereby showing that even in this simple case the non-local terms can induce instability that leads to the rod assuming helical shapes.Comment: 34 pages, 9 figure

The fundamentals of non-singular dislocations in the theory of gradient elasticity: dislocation loops and straight dislocations

The fundamental problem of non-singular dislocations in the framework of the theory of gradient elasticity is presented in this work. Gradient elasticity of Helmholtz type and bi-Helmholtz type are used. A general theory of non-singular dislocations is developed for linearly elastic, infinitely extended, homogeneous, and isotropic media. Dislocation loops and straight dislocations are investigated. Using the theory of gradient elasticity, the non-singular fields which are produced by arbitrary dislocation loops are given. `Modified' Mura, Peach-Koehler, and Burgers formulae are presented in the framework of gradient elasticity theory. These formulae are given in terms of an elementary function, which regularizes the classical expressions, obtained from the Green tensor of the Helmholtz-Navier equation and bi-Helmholtz-Navier equation. Using the mathematical method of Green's functions and the Fourier transform, exact, analytical, and non-singular solutions were found. The obtained dislocation fields are non-singular due to the regularization of the classical singular fields.Comment: 29 pages, to appear in: International Journal of Solids and Structure