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In this paper, stress gradient theory is used to model the static pull-in instability and size effect of electrostatic nanocantilevers in the presence of electrostatic and dispersion (Casimir/van der Waals) forces. The Differential transformation method (DTM) is employed to solve the nonlinear constitutive equation of the nanostructure as well as numerical methods. The basic engineering design parameters such as critical tip deflection and pull-in voltage of the nanostructure are computed. It is found that in the presence of dispersion forces, both pull-in voltage and deflection of the nanobeam increase with increasing the size effect. Compared to the pull-in voltage, the pullin deflection of the beam is less sensitive to the size effect at sub-micrometer scales. On the other hand, the size effect can increase the pull-in parameters of the nano-actuators only in sub-micrometer scales. The results indicate that the proposed analytical solutions are reliable for simulating nanostructures at sub-micrometer ranges.

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In this paper, the effect of size of an atomic force microscope (AFM) with an assembled cantilever probe (ACP) on resonant frequencies and their sensitivities are investigated using the strain gradient elasticity theory. The proposed ACP comprises a horizontal microcantilever, an extension and a tip located at the free end of the extension, which make the AFM capable of scanning the sample sidewall. First, the governing differential equation of motion and boundary conditions for dynamic analysis are obtained by a combination of the basic equations of the strain gradient elasticity theory and Hamilton principle. Afterwards, the flexural resonant frequency and sensitivity of the proposed AFM microcantilever are obtained numerically. The results of the proposed method are compared with those of modified couple stress and classical beam theories. The comparison shows that the difference between the results predicted by the strain gradient elasticity theory with those obtained by couple stress and classical beam theories become significant when the horizontal cantilever thickness comes approximately close to the material length scale parameter.

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In this paper, the anti-plane stress analysis in an infinite elastic plane with multiple cracks is carried out by using the distributed dislocation technique. The solution is obtained for an infinite plane containing the screw dislocation via Fourier transform of biharmonic equation for the analysis of infinite plane in gradient elasticity. These solutions are used to perform integral equations for an infinite plane weakened by multiple straight cracks. Integral equations are hypersingular type which are solved numerically for density of dislocation on the cracks surfaces. The numerical method in Chebyshev series form are used to solve the hypersingular integral equations. The solution of integral equations leads to dislocation density functions. The stress intensity factor for cracks tips are formulated in terms of density of dislocation. Employing the definition of dislocation density, stress intensity factors for cracks tips are calculated. The influence of size-effect and crack location on the stress intensity factors are studied. To confirm the validity of formulations, numerical values of stress intensity factors are compared with the results in the literature. The results of the present approach are in excellent agreement with those in the literature. Some new examples with different geometrics of cracks are solved to illustrate the applicability of procedure.

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When the cantilever beam thickness is scaled down to micron, the dimension of material and the intrinsic length scale affect the mechanical behavior of the beam. The purpose of this paper is analyzing the bending of cantilever micro-beam and presenting an exact relation for the beam deflection using Chen-Wang gradient plasticity theory. To this end, the Euler-Bernoulli beam theory is utilized to model a micro-beam and three cases including elastic, rigid-plastic and elasto-plastic beams are considered. Clear relations for elastic and plastic strains are given. For all mentioned cases, the beam deflection is determined for different intrinsic lengths and the obtained results are compared with each other and the data obtained from experimental tests and some explanations are presented. The results obtained from classical theory are also shown in the results section to prove that classical theories don’t have the capability to predict behavior of micron-size structures precisely. Numerical results clarify the dependence of responses to the range of dimensions and intrinsic lengths. The comparison between present results and those observed from experimental tests authenticate the reliability of utilized gradient theory.

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In recent decade, modeling the instability of nanostructures has attracted many attentions in nanomechanics. Nanomechanical switches are fundamental building blocks for the design of NEMS applications, such as nanotweezers and nanoscale actuators. One common type of NEMS including nano-bridge in micro mirrors is used. At nano-scales, the decreasing gap between the two electrodes makes surface traction due to molecular interaction such as van der Waals that must be taken into account in the analysis of NEMS. In this study, strain gradient theory has been used to investigate the size dependent pull-in instability of beam-type (NEMS)where is an inherent instability in them. The von-Karman nonlinear strain has been applied to derive the constitutive equation of the system. Effect of intermolecular force have been included in the nonlinear governing equations of the systems. Homotopy perturbation method (HPM) has been employed to solve the nonlinear equations. Effect of intermolecular attraction and the size dependency and the importance of coupling between them on the instability performance i.e. critical deflection and instability voltage have been discussed. According the findings of this research, one can conclude that intermolecular forces decrease pull-in voltage and size effect parameter in nano scale leads to increase of pull-in parameters. Also HPM method can be applied as efficient method to analyze beam type nano structures.

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In this paper, pull-in instability of a cantilever beam type nanoactuator made of the functionally graded material (FGM) based on higher order modified strain gradient theory investigated. It is assumed that the functionally graded beam, made of germanium and silicon, follows the volume fraction definition and law of mixtures, and its properties change as a power function through its thickness. By changing the germanium constituent volume fraction percent of the nano-beam, five different types of the nano-beams are investigated. The influences of the volume fraction index, length scale parameter and the intermolecular forces, on the pull-in instability are examined. Principle of minimum total potential energy used to derive the nonlinear governing differential equation and consistent boundary conditions which is then solved using the differential quadrature method (DQM). The present analysis is validated through direct comparisons with published other research methods and experimental results and after comparison excellent agreement has been achieved between new solution method and other experimental and numerical solution results. Besides, the results demonstrate that size effect and amount of volume fraction have a substantial impact on the pull-in instability behavior of beam-type nanoactuator.

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In this paper, free vibration of rotating microbeams based on the strain gradient theory and Euler-Bernoulli beam assumptions is investigated. The Hamilton's Principle is applied on the attained strain and kinetic energy relations to obtain the equations of motion for the rotating microbeam. Then, by employment of the adimensional parameters, the nondimensional form of the equations of motion is derived. By applying the Galerkin approach on the dynamic equations of motion, the flapping and axial natural frequencies are calculated. Subsequently, the current results are validated by the existed papers results. After validation of the present results, the effects of the thickness to the material length scale parameter ratio, rotation speed and Poisson's coefficient on the flapping and axial frequencies are studied and the strain gradient theory results are compared with the modified couple stress and classical theories. The results show that the type of the theory which is appointed has essential effects on the predicted natural frequencies. The effect of rotation speed on the possibility of the occurrence of internal resonances is also examined. In addition, for the first time, the effect of different mentioned theories on the axial natural frequencies are inspected. The presented results illustrated, by considering the strain gradient theory, varying the Poisson's coefficient changes the axial frequencies, while, the modified couple stress and classical theories are incompetent to predict any variations on the axial frequencies and the mentioned theories predict the same results for axial frequencies.

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In this paper, the dynamic behavior of atomic force microscope (AFM) based on non-classical strain gradient theory was analyzed. For this aim atomic force microscope micro-beam with attached tip has been modeled as a lumped mass. Micro-beam has stimulated via a piezoelectric element attached to the end of clamped and non-linear partial differential equation of the system has extracted based on Euler-Bernoulli theory and to be converted into ordinary differential equation by using Galerkin and separation method. The classic continuum theory because of lack of consideration size effect that has been observed in many experimental studies, has little accuracy in predicting the mechanical behavior of Nano devices. In this study, the stability region of micro-beam are determined analytically and validated by comparison with numerical results. Difference between presented analysis in dynamic behavior of micro-beam by classic and non-classic theories has been shown with variety of diagrams. It is clear that consideration the size effect changes the dynamical behavior of the problem completely and it is possible while classical theory predicts stable behavior for microscope the size effect is caused bi-stability. The results in this paper are very useful for the design and analysis of atomic force microscope.

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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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ٍExperimental studies indicates that the mechanical behavior of materials at micro and nano scales are size-dependent. Since the classical continuum mechanics theories cannot capture the size effect, employment of different non-classical theories has received a considerable attention among researchers. In this study, the finite element formulation is presented to investigate the bending of square microplates with circular hole subjected to uniform pressure based on the three-dimensional strain gradient elasticity theory. For this account, the 8-node C^1 continuous hexahedral element is introduced in which, in addition to the values of displacement components, some related higher-order mix derivatives are further considered as nodal values. The governing equations are derived based on the strain gradient theory and three-dimensional elasticity model and the finite element formulation is presented using the introduced element. Note that by considering some specified values for coefficients of strain gradient theory, the numerical results can be obtained for modified strain gradient theory and modified couple stress theory. To demonstrate the efficiency of the proposed finite element, the convergence and accuracy of the results are firstly checked and then the impacts of geometrical parameters on the bending of microplates with circular hole are studied.

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