Sharif Digital Repository

Incorporation of the first gradient of strain, in addition to the strain itself, into the strain energy density of an elastic solid leads to Mindlin's first strain gradient theory, which is useful for examination of size effect as well as other mechanical phenomena at the nano-scale. For isotropic elastic solids, the first strain gradient theory, in addition to the two independent Lamé constants, gives rise to five new material constants which in turn reduce to two material parameters, ℓ1 and ℓ2 with dimension of length. The evaluation of these parameters, however, has posed serious challenges, both experimentally and theoretically. In this work ab initio method is used to compute the... 

This study investigates influence of fringing field effect on the voltage dependent behavior of electrostatically actuated micro-beams. For this purpose, the size dependent beam model is used. Strain gradient formulation is utilized to consider size effects. The effect of fringing field effect on the beam's behavior is investigated and it is shown that lack of considering the fringing field effect in the formulation of the problem may lead to considerable error in predicting the size dependent micro-beams behavior under the effect of electrostatic actuation. The results of this research can be used for safe and stable design of electrostatically actuated micro-beams  

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  

Based on nonlocal strain gradient theory (NSGT), transient behavior of a porous functionally graded (FG) nanoplate due to various impulse loads has been studied. The porous nanoplate has evenly and unevenly distributed pores inside its material structure. Impulse point loads are considered to be rectangular, triangular and sinusoidal types. These impulse loads lead to transient vibration of the nanoplate which is not studied before. NSGT introduces a nonlocal coefficient together with a strain gradient coefficient to characterize small size influences due to non-uniform stress and strain fields. Galerkin's approach has been performed to solve the governing equations and also inverse Laplace... 

This paper presents a nonlocal strain gradient theory for capturing size effects in buckling analysis of Euler-Bernoulli nanobeams made of threedimensional functionally graded materials. The material properties vary according to any function. These models can degenerate to the classical models if the material length-scale parameters is assumed to be zero. The Hamilton's principle applied to drive the governing equation and boundary conditions. Generalized differential quadrature method used to solve the governing equation. The effects of some parameters, such as small-scale parameters and constant material parameters are studied. © 2022 PAGEPress Publications. All rights reserved  

In this paper, vibration suppression of micro-or nano-scale beams subjected to nonlinear distributed electrostatic force is studied. For the sake of precision, we use the beam model derived from strain gradient elasticity theory aimed at prediction of size effect. In addition, the electrostatic force is considered with first order fringing field correction. The continuous model of the strain gradient beam is truncated by using Kantorovich method as a semi-analytical finite element method. A boundary control feedback law is proposed to suppress forced vibrations of the beam. Both measurements and actuations are taken place in the boundary to avoid spillover instabilities. Simulation results... 

In this paper, the mechanical behavior of micro-rotating disks is investigated utilizing the strain gradient theory. The governing equation and boundary conditions are derived utilizing the variational method. The analytical solution for the derived equation is also presented. As a case study, some numerical results are presented to emphasize the importance of utilization of non-classical theories such as the strain gradient elasticity instead of the classical continuum theory in dealing with micro-rotating disks  

A screw dislocation outside an infinite cylindrical nano-inhomogeneity of circular cross section is considered within the isotropic theory of gradient elasticity. Fields of total displacements, elastic and plastic distortions, elastic strains and stresses are derived and analyzed in detail. In contrast with the case of classical elasticity, the gradient solutions are shown to possess no singularities at the dislocation line. Moreover, all stress components are continuous and smooth at the interface unlike the classical solution. As a result, the image force exerted on the dislocation due to the differences in elastic and gradient constants of the matrix and inhomogeneity, remains finite when... 

Conventional continuum theory does not account for contributions from length scale effects which are important in modeling of nano-beams. Failure to include size-dependent contributions can lead to underestimates of deflection, stresses, and pull-in voltage of electrostatic actuated micro and nano-beams. This research aims to use nonlocal and strain gradient elasticity theories to study the static behavior of electrically actuated micro- and nano-beams. To solve the boundary value nonlinear differential equations, analogue equation and Gauss–Seidel iteration methods are used. Both clamped-free and clamped–clamped micro- and nano-beams under electrostatical actuation are considered where... 

In this paper, strain gradient thermo-elasticity formulation for axisymmetric Functionally Graded (FG) thick-walled cylinders is presented. For this purpose, the elastic strain energy density function is considered to be a function of gradient of strain tensor in addition to the strain tensor. The material properties are assumed to vary according to a power law in radial direction. Using the constitutive equations and equation of equilibrium in the cylindrical coordinates, a fourth order non-homogenous governing equation for thermo-elastic analysis of thick-walled FG cylinders subjected to thermal and mechanical loadings is obtained and solved numerically. Results show that the intrinsic... 

In this paper, a size-dependent formulation is presented for vibration analysis of micro-end mill tool. The formulation is developed based on the strain gradient elasticity theory in order to enhance the modeling capability of micro-size structures. Due to stubby geometry of micro-tool, the shear deformation and rotary inertia effects are considered in the derivation of equations. Hence, based on the strain gradient Timoshenko beam theory, the extended Hamilton's principle is used to formulate a detailed dynamical model of the rotating micro-tool. The dynamical model includes a set of partial differential equations with gyroscopic coupling produced due to the spindle rotation. The governing... 

The small-scale effects on the mechanical responses of micro-rotating disks with angular acceleration are investigated based on the strain gradient theory, as one of the powerful non-classical continuum theories which have been developed to justify the empirical observations of mechanical behavior in small-scale structures and components. The differential equations governing motion of the micro-disk elements in radial and circumferential direction together with the corresponding boundary conditions are derived. Then, an analytical solution is presented for the components of the displacement field which can be used as a base for determination of the components of the stress field. In a... 

During the varying angular speed timespans of the start or shutdown of rotating machinery, the machinery components may be subjected to intense mechanical loadings which should be taken into account by its fabricator in the designing processes. In the microscale rotating systems, where the angular velocity is typically very high, the importance of this issue is much higher. In this paper, a comprehensive strain-gradient elasticity formulation is presented for functionally graded rotating micro-disks under the effects of varying angular velocity. The gradation of the constituent material along the radial direction can be a helpful option to mitigate the stresses in rotating micro-disks under... 

The classical continuum theory not only underestimates the stiffness of microscale structures such as microbeams but is also unable to capture the size dependency, a phenomenon observed in these structures. Hence, the non-classical continuum theories such as the strain gradient elasticity have been developed. In this paper, a Timoshenko beam finite element is developed based on the strain gradient theory and employed to evaluate the mechanical behavior of microbeams used in microelectromechanical systems. The new beam element is a comprehensive beam element that recovers the formulations of strain gradient Euler–Bernoulli beam element, modified couple stress (another non-classical theory)... 

Mindlin's (1965) second strain gradient theory due to its competency in capturing the effects of edges, corners, and surfaces is of particular interest. Formulation in this framework, in addition to the usual Lamé constants, requires the knowledge of sixteen additional materials constants. To date, there are no successful experimental techniques for measuring these material parameters which reflect the discrete nature of matter. The present work gives an accurate remedy for the atomistic calculations of these parameters by utilizing the first principles density functional theory (DFT) for the calculations of the atomic force constants combined with an analytical formulation. It will be shown...