The increasing number of nano and microscopic structural devices and their importance in several technological fields have pushed the research towards the formulation of mathematical models suitable for capturing mechanical small-scale effects. Usually, in these cases, an accurate modeling requires the simulation of the microstructure, the reproduction of intermolecular interactions and heterogeneity at the micro/nanoscale. Therefore, for this kind of mechanical modeling, classical local continuum theories fail in reproducing the real behavior at small-scale due to the inability to reproduce these scale-dependent effects. For this reason, an advanced theory is developed in the recent decades. Such formulation is known as nonlocal theory and it is able to take into account nonlocal effects. Among these nonlocal effects, there are long-range interactions, size-effects and heterogeneity of the material, strain localizations, and so on. Usually, in these advanced models, the nonlocal effects are reproduced by means of some additional terms in the governing equations. There are several kind of nonlocal models provided in literature. Among these various models, this chapter considers the displacement based nonlocal models which belong to the mechanically based nonlocality. Following this approach, the nonlocal effects are modeled as additional body forces acting on material volumes depending on their relative displacements. An overview of the main results of this theory and a summary of the other nonlocal models are reported in this chapter showing their differences and likeness.
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