Free and Parametric Vibrations of Cylindrical Shells under Static and Periodic Axial Loads

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On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory

In the recent literature stance, purely nonlocal theory of elasticity is recognized to lead to ill-posed problems. Yet, we show that, for a beam, a meaningful energy bounded solution of the purely nonlocal theory may still be defined as the limit solution of the two-phase nonlocal theory. For this, we consider the problem of free vibrations of a flexural beam under the two-phase theory of nonlocal elasticity with an exponential kernel, in the presence of rotational inertia. After recasting the integro-differential governing equation and the boundary conditions into purely differential form, a singularly perturbed problem is met that is associated with a pair of end boundary layers. A multi-parametric asymptotic solution in terms of size-effect and local fraction is presented for the eigenfrequencies as well as for the eigenforms for a variety of boundary conditions. It is found that, for simply supported end, the weakest boundary layer is formed and, surprisingly, rotational inertia affects the eigenfrequencies only in the classical sense. Conversely, clamped and free end conditions bring a strong boundary layer and eigenfrequencies are heavily affected by rotational inertia, even for the lowest mode, in a manner opposite to that brought by nonlocality. Remarkably, all asymptotic solutions admit a well defined and energy bounded limit as the local fraction vanishes and the purely nonlocal model is retrieved. Therefore, we may define this limiting case as the proper solution of the purely nonlocal model for a beam. Finally, numerical results support the accuracy of the proposed asymptotic approach

Free Vibrations of a Non-uniformly Heated Viscoelastic Cylindrical Shell

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Modeling pull-in instability of CNT nanotweezers under electrostatic and van der Waals attractions based on the nonlocal theory of elasticity

This work investigates the electromechanical response and pull-in instability of an electrostatically-actuated CNT tweezer taking into consideration a TPNL constitutive behavior of the CNTs as well as the intermolecular forces, both of which provide a significant contribution at the nanoscale. The nonlocal response of the material introduces two additional parameters in the formulation, which are effective in capturing the size effects observed at the nanoscale. The problem is governed by a nonlinear integrodifferential equation, which can be reduced to a sixth-order nonlinear ODE with two additional boundary conditions accounting for the nonlocal effects near to the CNT edges. A simplified model of the device is proposed based on the assumption of a linear or parabolic distribution of the loading acting on the CNTs. This assumption allows us to formulate the problem in terms of a linear ODE subject to two-point boundary conditions, which can be solved analytically. The results are interesting for MEMS and NEMS design. They show that strong coupling occurs between the intermolecular forces and the characteristic material lengths as smaller structure sizes are considered. Considering the influence of the nonlocal constitutive behavior and intermolecular forces in CNT tweezers will equip these devices with reliability and functional sensitivity, as required for modern engineering applications

Local Buckling of Composite Laminated Cylindrical Shells with Oblique Edges under External Pressure: Asymptotic and Finite Element Simulations

The problem of local buckling ofa thin composite laminated cylindrical shell under external pressure is studied. Each layer of the shell is assumed to be isotropic. The special case of the shell being non-circular and/or having no plane edges is considered here. Presupposing that buckling takes place in the neighborhood of some so-called “weakest” generator, the asymptotic Tovstik’s method is appliedfinding the critical pressure and the elgenmodes. As an example, buckling of a three-layered circular thin cylinder with a sloped edge is investigated. Besides the asymptotic approach the finite element simulation is applied to facilitate the estimation of the range to which the results obtained can be applied

Pull-in Instability Analysis of a Nanocantilever Based on the ‎Two-Phase Nonlocal Theory of Elasticity

This paper deals with the pull-in instability of cantilever nano-switches subjected to electrostatic and intermolecular forces in the framework of the two-phase nonlocal theory of elasticity. The problem is governed by a nonlinear integro-differential equation accounting for the external forces and nonlocal effects. Assuming the Helmholtz kernel in the constitutive equation, we reduce the original integro-differential equation to a sixth-order differential one and derive a pair of additional boundary conditions. Aiming to obtain a closed-form solution of the boundary-value problem and to estimate the critical intermolecular forces and pull-in voltage, we approximate the resultant lateral force by a linear or quadratic function of the axial coordinate. The pull-in behavior of a freestanding nanocantilever as well as its instability under application of a critical voltage versus the local model fraction are examined within two models of the load distribution. It is shown that the critical voltages calculated in the framework of the two-phase nonlocal theory of elasticity are in very good agreement with the available data of atomistic simulation

Parametric Vibrations of Viscoelastic Cylindrical Shell under Static and Periodic Axial Loads

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Free vibrations of nonlocally elastic rods

Several of the Eringen’s nonlocal stress models, including two-phase and purely nonlocal integral models, along with the simplified differential model, are studied in case of free longitudinal vibrations of a nanorod, for various types of boundary conditions. Assuming the exponential attenuation kernel in the nonlocal integral models, the integro-differential equation corresponding to the two-phase nonlocal model is reduced to a fourth order differential equation with additional boundary conditions taking into account nonlocal effects in the neighbourhood of the rod ends. Exact analytical and asymptotic solutions of boundary-value problems are constructed. Formulas for natural frequencies and associated modes found in the framework of the purely nonlocal model and its ”equivalent” differential analogue are also compared. A detailed analysis of solutions suggests that the purely nonlocal and differential models lead to ill-posed problems

Thermoparametric Vibrations of Noncircular Cylindrical Shell in Nonstationary Temperature Field

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