Surface effect and nonlocal effect are incorporated into classical Euler beam equation to study the static bending behavior of nanobridges. The generalized Young-Laplace equation and core-shell model are used to model the surface effect. The nonlocal effect is introduced in bending moment equation. Results show that a positive surface tension causes the nanobridge more difficult to bend while the nonlocal effect causes the nanobridge easier to bend. Moreover, the influences of the surface effect and the nonlocal effect on the nanobridge bending behavior are found to be dependent of the applied force position. This study indicates the importance of the excitation force position in the characterizations and applications of nanobridges.
In this paper, the constitutive relations of nonlocal elasticity theory are presented for application in the analysis of carbon nanotubes (CNTs) when modelled as Euler-Bernoulli beams, Timoshenko beams or as cylindrical shells. In particular, the shear stress and strain relation for the nonlocal Timoshenko beam theory is discussed in great detail due to a misconception by some researchers that the nonlocal effect should appear in this constitutive relation. Different theories for proposing the value of the small scale parameter are also introduced and a recommendation for the value from the standpoint of wave propagation of CNTs is given.
In this paper we prove that every nonlinear ∗ -Lie derivation from a factor von Neumann algebra into itself is an additive ∗ -derivation.