The conventional modified couple stress theory cannot model the correct behavior of the longitudinal dispersion and acts the same as the classical theory in the face of such problems. In this paper, the micro-inertia-based couple stress theory is used to triumph over this deficiency. The developed theory is imposed to tackle the longitudinal dispersion of aluminum beams in two distinct scales. Convenient available experimental data obtained for a macro-scale aluminum rod and aluminum crystals are utilized to determine the corresponding micro-inertia length scale parameters and show the scale-dependent nature of this parameter for the first time. In addition, a higher order micro-rotation relation is employed to describe the higher order micro-inertia effects. This relation leads to a developed equation of motion containing an additional term compared with the first-order relation. The obtained results indicate that only higher order micro-inertia effect that is proposed in this study for the first time is able to capture the highly nonlinear behavior of dispersion curves (including an extremum/inflection point), which has experimentally been observed for phonons propagating in the longitudinal direction in an aluminum crystal.
This paper presents a size‐dependent analysis of carbon nanostructures to put forward some new insights into the conventional modified couple stress theory. In fact, a micro‐inertia length scale parameter is added to the equation of motion to compensate the deficiency of the theory in describing the size effect of the longitudinal dispersion. In this study, the longitudinal dispersions of carbon nanotubes, graphene, and graphite layers are investigated to obtain the micro‐inertia length scale parameter, and the flexural dispersions of carbon nanotubes and graphene layer are assessed to expose the size dependency of the couple‐stress length scale parameter. The results indicate that the micro‐inertia length scale parameter can directly be related to the lattice size, while the couple‐stress length scale parameter depends on some more factors like geometry, size, and material properties. Finally, some general relations are proposed for the first time to make a connection between these two distinct parameters and other features.
In this paper, an appropriate and accurate algorithm is pro- posed to diagnosis of lateral or vertical cracks on beam, based on beam natural frequencies. Clamped-free boundary conditions are assumed for the beam. The crack in beam is modelled by without mass torsion spring. Then, the relationship between the beam natural frequencies, location and stiffness of the crack is presented by using the Rayleigh quotient and the governing equation is solved by using generalized differential quadrature method (GDQM). If there is only one crack in the beam, then three natural frequencies are used as inputs to the algorithm and mode shapes corresponding to each the natural frequencies are calculated. Finally, type, location and severity of cracks in beam, are diagnosed.
In this paper, dynamic analysis of a cantilever beam with micro-scale dimensions is presented. The micro-cantilever is subjected to harmonic base excitation and constant force at micro-cantilever tip. By Euler-Bernoulli beam theory assumptions, the mathematical formulation of vibrating micro-cantilever beam is derived using extended Hamilton principle. The governing partial-diffrential equation is solved by reconstruction of variational iteration method (RVIM), with possession of its boundary conditions. The RVIM is an approximate method of solving that answers easy and quick and has high accuracy.
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