The fractal model of non-local elasticitywith long-range interactions

The mechanically-based model of non-local elasticity with long-range interactions is framed, in thisstudy, in a fractal mechanics context. Non-local interactions are modelled introducing long-range centralbody forces between non-adjacent volume elements of the elastic continuum. Such long-range interactionsare modelled as proportional to the product of interacting volumes, to the relative displacements ofthe centroids and to a distance-decaying function that is monotonically-decreasing with the distance.The choice of the decaying function is a key aspect of the model and it has been proved that anycontinuous function, strictly positive, is thermodynamically consistent and it leads to a material thatsatisfy the Drucker stability criterion [2]. Such a mathematical model of non-local elasticity has aninteresting mechanical counterpart that is described by a point-spring network with multiple springs withdistance-decaying stiffness.As the functional class of the distance-decaying function is modelled as a power-law function ofthe distance of interacting particles, then, in the 1D case, the governing operators are Marchaud-typefractional derivatives as proved by the authors in previous studies [1].In this study we aim to show that, as we assume that the stiffness associated to long-range interactionsis modelled as a self-similar transformation of the Euclidean distance with anomalous and real scalingexponent, the mechanical model of the non-local elasticity is a self-similar fractal object.In more detail, assuming a non-integer power-law decay of the long-range forces between adjacentvolumes of an ideal next nearest (NN) model, the scaling law of the stiffness of the long-range bonds isreadily obtained. The Hausdorff-Besitckovich (HB) fractal dimension provides the appropriate boundsof the decay coefficient necessary to maintain the self-similarity of the obtained fractal set. The NNmodel, however leads to mathematically inconsistent governing operator for general class of continuousdisplacement function and it is proved that in this case only one integer value of the long-range forcedecay is admissible leading to classical second-order differential operators.A different scenario is involved as we introduce, on mechanical grounds, the long-range interactionconcept so that as we refine observation scale, the interactions between particles is still involving thepresence of all the new, non-adjacent particles so that the original NN lattice is turned into a more refinedand realistic next to the nearest next (NNN) lattice model. Such a model is equivalent to the mechanicalmodel of the long-range interactions introduced by the authors to describe non-local elasticity. The modelis constituted of self-similar copies of elastic chains and henceforth it may be considered as a mechanicalfractal as we assume an unbounded domain. This fractal set is not coalescing with usual fractals since itretains all the informations of previous observation scales and henceforth it has been dubbed as multiscalefractal. In this context the HB dimension of the mechanical fractal may be obtained as a function of thedecaying exponent of the long-range interactions and it may be proved that the governing equation of1the problem are Marchaud fractional-type differential operator as already postulated by the authors in aprevious study [1]. Some conclusions about the use of fractional operators in the context of multiscaleapproach to non-local mechanics may be also withdrawn from previous considerations.