The mechanicallybased model of nonlocal elasticity with longrange interactions is framed, in this study, in a fractal mechanics context. Nonlocal interactions are modelled introducing longrange central body forces between nonadjacent volume elements of the elastic continuum. Such longrange interactions are modelled as proportional to the product of interacting volumes, to the relative displacements of the centroids and to a distancedecaying function that is monotonicallydecreasing with the distance. The choice of the decaying function is a key aspect of the model and it has been proved that any continuous function, strictly positive, is thermodynamically consistent and it leads to a material that satisfy the Drucker stability criterion [2]. Such a mathematical model of nonlocal elasticity has an interesting mechanical counterpart that is described by a pointspring network with multiple springs with distancedecaying stiffness. As the functional class of the distancedecaying function is modelled as a powerlaw function of the distance of interacting particles, then, in the 1D case, the governing operators are Marchaudtype fractional 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 longrange interactions is modelled as a selfsimilar transformation of the Euclidean distance with anomalous and real scaling exponent, the mechanical model of the nonlocal elasticity is a selfsimilar fractal object. In more detail, assuming a noninteger powerlaw decay of the longrange forces between adjacent volumes of an ideal next nearest (NN) model, the scaling law of the stiffness of the longrange bonds is readily obtained. The HausdorffBesitckovich (HB) fractal dimension provides the appropriate bounds of the decay coefficient necessary to maintain the selfsimilarity of the obtained fractal set. The NN model, however leads to mathematically inconsistent governing operator for general class of continuous displacement function and it is proved that in this case only one integer value of the longrange force decay is admissible leading to classical secondorder differential operators. A different scenario is involved as we introduce, on mechanical grounds, the longrange interaction concept so that as we refine observation scale, the interactions between particles is still involving the presence of all the new, nonadjacent particles so that the original NN lattice is turned into a more refined and realistic next to the nearest next (NNN) lattice model. Such a model is equivalent to the mechanical model of the longrange interactions introduced by the authors to describe nonlocal elasticity. The model is constituted of selfsimilar copies of elastic chains and henceforth it may be considered as a mechanical fractal as we assume an unbounded domain. This fractal set is not coalescing with usual fractals since it retains all the informations of previous observation scales and henceforth it has been dubbed as multiscale fractal. In this context the HB dimension of the mechanical fractal may be obtained as a function of the decaying exponent of the longrange interactions and it may be proved that the governing equation of 1 the problem are Marchaud fractionaltype differential operator as already postulated by the authors in a previous study [1]. Some conclusions about the use of fractional operators in the context of multiscale approach to nonlocal mechanics may be also withdrawn from previous considerations.
Di Paola, M., Zingales, M. (2010). The fractal model of nonlocal elasticity with longrange interactions. In ECCM2010. PARIGI : ECCOMAS.
The fractal model of nonlocal elasticity with longrange interactions
DI PAOLA, Mario;ZINGALES, Massimiliano
20100101
Abstract
The mechanicallybased model of nonlocal elasticity with longrange interactions is framed, in this study, in a fractal mechanics context. Nonlocal interactions are modelled introducing longrange central body forces between nonadjacent volume elements of the elastic continuum. Such longrange interactions are modelled as proportional to the product of interacting volumes, to the relative displacements of the centroids and to a distancedecaying function that is monotonicallydecreasing with the distance. The choice of the decaying function is a key aspect of the model and it has been proved that any continuous function, strictly positive, is thermodynamically consistent and it leads to a material that satisfy the Drucker stability criterion [2]. Such a mathematical model of nonlocal elasticity has an interesting mechanical counterpart that is described by a pointspring network with multiple springs with distancedecaying stiffness. As the functional class of the distancedecaying function is modelled as a powerlaw function of the distance of interacting particles, then, in the 1D case, the governing operators are Marchaudtype fractional 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 longrange interactions is modelled as a selfsimilar transformation of the Euclidean distance with anomalous and real scaling exponent, the mechanical model of the nonlocal elasticity is a selfsimilar fractal object. In more detail, assuming a noninteger powerlaw decay of the longrange forces between adjacent volumes of an ideal next nearest (NN) model, the scaling law of the stiffness of the longrange bonds is readily obtained. The HausdorffBesitckovich (HB) fractal dimension provides the appropriate bounds of the decay coefficient necessary to maintain the selfsimilarity of the obtained fractal set. The NN model, however leads to mathematically inconsistent governing operator for general class of continuous displacement function and it is proved that in this case only one integer value of the longrange force decay is admissible leading to classical secondorder differential operators. A different scenario is involved as we introduce, on mechanical grounds, the longrange interaction concept so that as we refine observation scale, the interactions between particles is still involving the presence of all the new, nonadjacent particles so that the original NN lattice is turned into a more refined and realistic next to the nearest next (NNN) lattice model. Such a model is equivalent to the mechanical model of the longrange interactions introduced by the authors to describe nonlocal elasticity. The model is constituted of selfsimilar copies of elastic chains and henceforth it may be considered as a mechanical fractal as we assume an unbounded domain. This fractal set is not coalescing with usual fractals since it retains all the informations of previous observation scales and henceforth it has been dubbed as multiscale fractal. In this context the HB dimension of the mechanical fractal may be obtained as a function of the decaying exponent of the longrange interactions and it may be proved that the governing equation of 1 the problem are Marchaud fractionaltype differential operator as already postulated by the authors in a previous study [1]. Some conclusions about the use of fractional operators in the context of multiscale approach to nonlocal mechanics may be also withdrawn from previous considerations.File  Dimensione  Formato  

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