Locality is then added to the created gap, and this triggers the nonlocal deflection to tend to the classic-local deflection.
However, that is not the case for the TPNI stress model, since its deflections monotonically evolve with respect to the nonlocal parameter in all the above explored problems. It is also critical to mention that the responses of the discrete lattice atomic models of a nanobeam appear to have a flexible behavior in comparison with the classical ones [82], unlike the responses of the strain gradient elasticity theory in one-dimensional problems [83–86].
As regards the free vibration problem, the eigenfrequencies of the nonlocal integral models appear to have a softening response in comparison with those of the classic- local model when the nonlocal parameter increases. Unlike the nonlocal differential model, no paradoxes are raised for the fundamental eigenfrequencies of the nonlocal integral models of a cantilever beam.
The responses of the nonlocal integral models differ from that of the nonlocal differential model as well. To be more specific, the eigenfrequencies of the TPNI stress model withk1=0.01 and those of the model of the modified kernel generally take smaller values than the eigenfrequencies of the nonlocal differential model.
By all the investigated models, the eigenfrequencies of the model of the modified kernel generally take the smallest values with respect to the nonlocal parameter. Need- less to say, the behavior of the fundamental eigenfrequency is noteworthy as well.
The aforementioned behavior is particularly affected by the equilibrium between the locality and the nonlocality, imposed by the model of the modified kernel on each problem that studied. Why is the fundamental eigenfrequency more sensitive to this phenomenon than the other eigenfrequencies? It still remains an unanswered question.
A kind of convergence of the numerical solutions, accomplished by successively finer meshes, also suggests that the FEM is stable for all the problems that studied.
As regards the static and the dynamical problems, the use of the modified kernel in a finite domain can be regarded the optimal selection for a mixed-type local/nonlocal model applied to the nanodevice’s design.
The overall conclusions drawn from this chapter are encouraging of triggering the study of complicated nanostructural problems such as nanoframes, nanoplates and nanoshells.
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