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The governing differential equation and both classical and non-classical boundary conditions of strain gradient bars are derived using variational approach. A closed-form analytical solution is obtained for static torsion and the characteristic equation, which gives the natural frequencies, is derived and analytically solved for the free torsional vibrations of the strain gradient microbars. A fixed-fixed microbar is considered as a specific case to investigate the torsional size-dependent static and free-vibration behavior of strain gradient microbars. The results of the current model are compared to those of the modified couple stress and classical theories  

The effect of surface and interface elasticity in analysis of the Saint-Venant torsion problem of an eccentrically two-phase circular nanobar is considered. The problem is formulated in the context of Gurtin's surface elasticity. For a rigorous solution of the proposed problem, conformal mapping together with a Laurent series expansion are employed. At the nanoscales the usual classical theories cease to hold and the corresponding results deteriorate. The numerical results well illustrate that the torsional rigidity of the mentioned nanosized structural elements are significantly affected by the size. Some applications of the given results can be contemplated in the design of micro/nano... 

The effect of surface and interface elasticity in the analysis of the Saint-Venant torsion problem of an eccentrically two-phase fcc circular nanorod is considered; description of the behavior of such a small structure via usual classical theories cease to hold. In this work, the problem is formulated in the context of the surface/interface elasticity. For a rigorous solution of the proposed problem, conformal mapping with a Laurent series expansion are employed together. The numerical results well illustrate that the torsional rigidity and stress distribution corresponding to such nanosized structural elements are significantly affected by the size. In order to employ surface and interface... 

The elastic behavior of a screw dislocation which is positioned inside the shell domain of an eccentric core-shell nanowire is addressed with taking into account the surface/interface stress effect. The complex potential function method in combination with the conformal mapping function is applied to solve the governing non-classical equations. The dislocation stress field and the image force acting on the dislocation are studied in detail and compared with those obtained within the classical theory of elasticity. It is shown that near the free outer surface and the inner core-shell interface, the non-classical solution for the stress field considerably differs from the classical one, while... 

In this paper, the scattering of anti-plane shear waves in an infinite matrix containing a multi-coated nanofiber/nanotube is studied. Based on the fact that the surface to volume ratio for nano-size objects increases, the usual classical theories which generally neglect the surface/interface effects fail to provide reasonable results. Therefore, to analyze the problem the wave-function expansion method is coupled with the surface/interface elasticity theory. In order to provide some quantitative results through consideration of several examples, the knowledge of the relevant surface and/or interface properties of the corresponding constituent materials are required. For this reason, part of... 

The classical continuum theory is neither able to accurately model the mechanical behavior of micro/nano-scale structures nor capable of justifying the size-dependent behavior observed in these structures; so the non-classical continuum theories such as the strain gradient theory have been emerged and developed. In order to enable the finite element method (FEM) to more accurately deal with the problems in micro/nano-scale structures, a size-dependent Euler-Bernoulli beam element is developed based on the strain gradient theory. Compared to the classical Euler-Bernoulli beam element, the nodal displacement vector of the new Euler-Bernoulli beam element has an additional component, i.e. the... 

This paper investigates the deflection and static pull-in voltage of microcantilevers based on the modified couple stress theory, a non-classic continuum theory capable to predict the size effects for structures in micron and sub-micron scales. It is shown that the couple stress theory can remove the gap between the experimental observations and the classical theory based simulations for the static pull-in voltage  

This paper investigates the deflection and static pull-in of microbridges based on the modified couple stress theory, a non-classic continuum theory able to predict the size effects for structures in micron and sub-micron scales. The beam is modeled using EulerBernoulli beam theory and the nonlinearities caused by mid-plane stretching have been considered. It is shown that modified couple stress theory predicts size dependent normalized deflection and pull-in voltage for microbeams while according to classical theory the normalized behavior of microbeams is independent of the size of the beam. According to results, when the thickness of the beam is in order of length scale of the beam... 

In this paper, utilizing the non-local theory, the resonant frequency and sensitivity of an AFM microcantilever are investigated. To that end, the governing equation of motion and corresponding boundary conditions are obtained using the variational approach. Afterwards, the resonant frequency and sensitivity of the AFM microcantilever are derived analytically and depicted in some figures versus the contact stiffness of the sampling surface. The results of the current model are compared to those of the classical theory. The comparison shows that the difference between the results of the non-local theory and those of the classical theory is significant when the non-local parameter is high but... 

We consider the quantum vacuum effects of the massless scalar fields that are non-minimally coupled to the background geometry of a collapsing homogeneous ball of dust. It is shown that for a definite range of coupling constants, there are repulsive quantum vacuum effects, capable of stopping the collapse process inside the black hole and precluding the formation of singularity. The final fate of the collapse will be a black hole with no singularity, inside which the matter stays balanced. The density of the final static matter will be close to the Planck density. We show that the largest possible radius of the stable static ball inside a black hole with Schwarzschild mass M is given by... 

Abstract: We consider the back reaction of the energy due to quantum fluctuation of the background fields considering the trace anomaly for Schwarzschild black hole. It is shown that it will result in modification of the horizon and also formation of an inner horizon. We show that the process of collapse of a thin shell stops before formation of the singularity at a radius slightly smaller than the inner horizon at the order of (Formula presented.). After the collapse stops the reverse process takes place. Thus we demonstrate that without turning on quantum gravity and just through the effects the coupling of field to gravity as trace anomaly of quantum fluctuations the formation of the... 

An exact solution is presented for the nonlinear cylindrical bending and postbuckling of shear deformable functionally graded plates in this paper. A simple power law function and the Mori-Tanaka scheme are used to model the through-the-thickness continuous gradual variation of the material properties. The von Karman nonlinear strains are used and then the nonlinear equilibrium equations and the relevant boundary conditions are obtained using Hamilton's principle. The Navier equations are reduced to a linear ordinary differential equation for transverse deflection with nonlinear boundary conditions, which can be solved by exact methods. Finally, by solving some numeral examples for simply... 

Near Horizon Extremal Geometries (NHEG) are solutions to gravity theories with SL(2, ℝ) × U(1) N (for some N) symmetry, are smooth geometries and have no event horizon, unlike black holes. Following the ideas by R. M. Wald, we derive laws of NHEG dynamics, the analogs of laws of black hole dynamics for the NHEG. Despite the absence of horizon in the NHEG, one may associate an entropy to the NHEG, as a Noether-Wald conserved charge. We work out "entropy" and "entropy perturbation" laws, which are respectively universal relations between conserved Noether charges corresponding to the NHEG and a system probing the NHEG. Our entropy law is closely related to Sen's entropy function. We also... 

Using the proposal for holographic entanglement entropy in higher derivative gravities, we compute holographic entanglement entropy for the conformal gravity in four dimensions which turns out to be finite. However, if one subtracts the contribution of the four dimensional Gauss-Bonnet term, the corresponding entanglement entropy has a divergent term and indeed restricted to an Einstein solution of the conformal gravity, the resultant entanglement entropy is exactly the same as that in the Einstein gravity. We will also make a comment on the first law of the entanglement thermodynamics for the conformal gravity in four dimensions  

The paper deals with scrutinizing the vibration and dynamic pull-in behaviours of carbon nanotubes with different dimensions and boundary conditions. The strain gradient theory as a nonclassical elasticity theorem is applied in this research to compensate the limitations of the classical theories in predicting the mechanical behaviours of the micro- and nanostructures in general and carbon nanotubes, in particular under electrostatic actuation. The results reveal that the mechanical properties of carbon nanotube (CNTs) computed using the strain gradient theory differ significantly with those obtained from the classical theory. The strain gradient theory leads to higher stiffness and pull-in... 

In this paper, static behavior and pull-in of micro tweezers is studied. The micro tweezer is modelled as two cantilever beams. Static behavior of the micro tweezer under the effect of electrostatic actuation is modelled using the Euler-Bernoulli beam theory. In order to capture size effects on the behavior of micro tweezers, modified couple stress theory is utilized. It is shown when the voltage between two electrodes increased from some specific value, micro beams adhere to each other and it is observed that the pull-in voltage predicted by the modified couple stress theory considerably differs with that of the classical theory of elasticity. Results of this paper can be used for accurate... 

In this paper, a nonlinear size-dependent Euler-Bernoulli beam model is developed based on a strain gradient theory, capable of capturing the size effect. Considering the mid-plane stretching as the source of the nonlinearity in the beam behavior, the governing nonlinear partial differential equation of motion and the corresponding classical and non-classical boundary conditions are determined using the variational method. As an example, the free-vibration response of hinged-hinged microbeams is derived analytically using the Method of Multiple Scales. Also, the nonlinear size-dependent static bending of hinged-hinged beams is evaluated numerically. The results of the new model are compared... 

The stress field of a screw dislocation inside an embedded nanowire is considered within the theory of strain-gradient elasticity. It is shown that the stress singularity is removed and all stress components are continuous and smooth across the interface, in contrast with the results obtained within the classical theory of elasticity. The maximum magnitude of dislocation stress depends greatly on the dislocation position, the nanowire size, and the ratios of shear moduli and gradient coefficients of the matrix and nanowire materials. © 2009 Acta Materialia Inc