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In this paper, the behavior of copper and steel rectangular plates with clamped boundary conditions subjected to underwater explosion loading is investigated. Cavitation is a phenomenon that occurs in this process. During the cavitation, the total pressure of the explosion becomes zero, so that the governing equations of motion time will be different before and after the cavitation. As a result, in terms of analysis and design, the cavitation time is significant in studying the behavior of a rectangular plate at underwater explosive loading. To calculate the cavitation time, the equations of motion of a rectangular plate underwater explosive loading are derived first, based on Hamilton principle and variation method. Then, in order to obtain the forced response of the rectangular plate, the exact free vibration solution of the rectangular plate is derived for exact mode shapes. Then, the speed and generated stress of plate during cavitation time are calculated and compared with the yield stress of copper and steel rectangular plates. Using this method, one can distinguish the cavitation with in the elastic or plastic regimes. Results show that the cavitation time is on the order of microsecond.

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The dynamic stability of single-walled carbon nanotubes (SWCNT) and double-walled carbon nanotubes (DWCNT) embedded in an elastic medium subjected to combined static and periodic axial loads are investigated using Floquet–Lyapunov theory and bounded solution theory. An elastic Euler- Bernoulli beam model is utilized in which the nested slender nanotubes are coupled with each other through the van der Waals (vdW) interlayer interaction. The Galerkin’s approximate method on the basis of trigonometric mode shape functions is applied to reduce the coupled governing partial differential equations to a system of the extended Mathieu-Hill equations. Applying Floquet–Lyapunov theory and Rung-Kutta numerical integration method with Gill coefficients, the influences of number of layer, elastic medium, exciting frequency and combination of exciting frequency on the instability conditions of SWCNTs and DWCNTs are investigated. A satisfactory agreement can be observed by comparison between the predicted results of Floquet–Lyapunov theory with bounded solutions theory ones. Based on results, increasing the number of layers, and elastic medium, dynamic stability of SWCNTs and DWCNTs surrounding elastic medium increase. Moreover, the instability of CNTs increases by increasing the exciting frequency.

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In this paper based on the Euler-Bernoulli beam model, the primary resonance a curved single carbon nanotube subjected to axial thermal force in the case of low temperature and high temperature and resting on a viscoelastic foundation is analytically investigated. The nonlinear partial differential governing equation is reduced to nonlinear ordinary differential governing equation by using of a single-mode Galerkin approximation along with the sinusoidal curvature for clamped-clamped single walled carbon nanotube under harmonic external force. The method of multiple scales is applied to determine the analytical primary resonance frequency response. Considering the curved geometry and the mid-plane stretching, a quadratic and cubic terms are presented in the governing equation. The effects of temperature change in high temperature and low temperature conditions, viscoelastic coefficients of medium, amplitude of sinusoidal curvature and excitation amplitude are investigated to study the property frequency response and development or elimination of forward and backward jumping phenomenon in primary resonance frequency response. The results show that these parameters have a significant effect on the frequency response of a curved single walled carbon nanaotubes under transvers harmonic force.

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