RANDOM VIBRATION OF NANOSTRUCTURES MODELED AS CONTINUOUS STRUCTURES WITH HIGHER ORDER NONLOCAL ELASTICITY

Due to discrete nature and the long-range force interactions, the mechanical response of nanostructures at a certain scale is influenced by a characteristic length at a lower scale. Being characterized by size effects and dispersion, the classical continuum theories are not adequate for the description of these lattice systems. The actual behaviour of these structures is captured by a continualization of the discrete dynamic equations (Metrikine & Askes, 2002) or by a mechanically-based approach to non-local elasticity (Di Paola et al., 2010). Recently, enriched theories, such as nonlocal integral/gradient elasticity theories, have been used for the modelling of nano-scale structures. The integral-type nonlocal models are based on constitutive laws at a point of a continuum that involve weighted averages of a state variable over a certain neighbourhood of that point, in disagreement with the principle of local action of the classical continuum mechanics. The gradient-type nonlocal models, while adhering to this principle mathematically, takes the field in the immediate vicinity of the point into account by enriching the local constitutive relations with the first or higher gradients of some state variables. In this paper, a higher integral nonlocal elasticity continuum model is proposed for modelling the nanostructures. Starting from the definition of the first order nonlocal strain tensor given by Eringen (1983), as the weighted average of the classical local strain tensor by means of a first order attenuation function, the proposed nonlocal model is based on the definition of higher order nonlocal strain tensors, which are the weighted averages of the local strain tensor by means of a higher order attenuation functions. The latter are the convolutions of the first order attenuation function and, in addition, the higher order strain tensors are strictly related to the first order one. The nonlocal elasticity constitutive model expresses the nonlocal stress tensor as a linear combination of the nonlocal strain tensors up to a given order. The coefficients of the linear combination are additional material parameters. By choosing a particular form of the first order attenuation function, the proposed nonlocal integral model is related to an equivalent gradient-type one, that involves the Helmholtz differential operator and its powers. The model is particularized in one-dimension, in order to search closed form solutions. Suitable non standard boundary conditions are determined to guarantee the equivalence between the integral and the gradient approaches. Dynamic equation of a beam with nonlocal elasticity behaviour under axial deformation is derived and the random vibrations are studied, in comparison with a discrete model.