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Size dependent behavior of materials appears for a structure when the characteristic size such as thickness or diameter is close to its internal length-scale parameter. In these cases, ignoring this behavior in modeling may leads to incorrect results. In this paper, strong effects of the size dependence on the static and dynamic behavior of the electrostatically actuated micro-beams have been studied. The equilibrium positions or fixed points of the gold and nickel micro-beams have been determined and shown that for a given DC voltage, there is a considerable difference between the fixed points gained using the classic beam theory and the modified couple stress theory. In addition, it has been shown that the static and dynamic pull-in voltages gained using the couple stress theory are several times higher than those gained using the classic beam theory. Some previous studies have applied the classic beam theory in their models and introduced a considerable hypothetical value of residual stress to match their experimental and incorrect theoretical results. It has been shown that using the modified couple stress theory decreases considerably the difference with the experimental results.

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Topology optimization of structure seeks to achieve the best material distribution in the Pre-determined design domain. In this paper, the effect of design parameters contains length scale parameter and evolutionary volume ratio in improved bi-directional evolutionary structural optimization method with soft kill approach is discussed. The main aim of this method is searching for the stiffest structure with a given volume of material using finite element method. At each iteration of finite element analysis, sensitivity number is calculated for each individual element in design domain and then converted to the nodal sensitivity number. With Filter Scheme and using length scale, an improved sensitivity numbers is defined. This number is used as a criterion for rating each element in design domain and determining the addition and elimination (remove) of elements. To increase the convergence of the optimization process, the accuracy of the new elemental sensitivity numbers is improved by considering the sensitivity history. This method is convergent and mesh-independent and there are no checkerboard pattern and local solutions in optimal topologies. Using three design samples, a cantilever and classical beam and Michell type structure, affecting factors will be discussed on the final design of the structure. Change of length scale parameter makes various schemes in final structures in which with increasing this parameter, more iteration is needed for convergent solution. Reducing evolutionary volume ratio forms different and even asymmetric topologies. Better final topologies are obtained with higher evolutionary volume ratios.

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In this study, the hydrostatic vibration analysis of an isotropic rectangular microplate in partial contact with a bounded fluid is studied. Modified couple stress theory based on the Kirchhoff plate assumptions are used to mathematically model the problem. The extended Hamilton’s principle is employed to drive the governing differential equation of motion and the corresponding boundary conditions. The transverse displacement of the microplate is approximated by a set of admissible functions which must satisfy the geometric boundary conditions. The fluid is assumed to be incompressible, inviscid and irrotational and the fluid velocity potential is obtained using the boundary and compatibility conditions. Natural frequencies of the microplate are calculated using the Rayleigh-Ritz method. To validate the present results, the natural frequencies of an isotropic macroplate in contact with fluid are compared with the available data in the literature and very good agreements are observed. Finally using the numerical data, the effect of different parameters such as thickness to length scale parameter, aspect ratio, length to thickness ratio and boundary conditions on the natural frequencies of the microplate are discussed in detail. We have observed that the difference between the natural frequencies predicted using the classical theory and the one evaluated by the modified couple stress theory is significant when thickness of the microplate is small, but diminishes as thickness increases.

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In this study, nonlinear behavior of an atomic force microscopes (AFM) immersed in acetone, water, carbon tetrachloride (CCl4), and 1-butanol is investigated using non-classical strain gradient theory. In this theory, the size effect of system is taking into account by means of material length scale parameter. The nonlinear behavior of the AFM is due to the nonlinearity of the AFM tip–sample interaction caused by the Van der Waals attraction/repulsion force. Behavior of micro beam immersed in liquid is completely different with its behavior in air and vacuum due to the existence of hydrodynamic force. The Resonant frequencies, mode shapes, governing nonlinear partial and ordinary differential equations (PDE and ODE) of motion, stability analysis, boundary conditions, potential function and phase-plane of the system are obtained analytically in the present study. Furthermore, the results are compared with the one obtained by the modified couple stress theory. For this purpose, the AFM and the probe at the free end of micro beam are modeled as a lumped mass. The fixed end of micro beam is excited by piezoelectric element. The nonlinear PDE of motion is derived based on Euler-Bernoulli theory by employing the Hamilton principle. The Galerkin method is utilized to gain the governing nonlinear ODE of motion and the obtained ODE is analytically solved by means of perturbation techniques.

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