Nonlocal Stress Field Research Articles

Buckling analysis of new quasi-3D FG nanobeams based on nonlocal strain gradient elasticity theory and variable length scale parameter

A size-dependent novel hyperbolic shear deformation theory of simply supported functionally graded beams is presented in the frame work of the non-local strain gradient theory, in which the stress accounts for only the nonlocal strain gradients stress field. The thickness stretching effect (&#949z ≠ 0) is also considered here. Elastic coefficients and length scale parameter are assumed to vary in the thickness direction of functionally graded beams according to power-law form. The governing equations are derived using the Hamilton principle. The closed-form solutions for exact critical buckling loads of nonlocal strain gradient functionally graded beams are obtained using Navier's method. The derived results are compared with those of strain gradient theory.

Nonlocal strain gradient-based vibration analysis of embedded curved porous piezoelectric nano-beams in thermal environment

This disquisition proposes a nonlocal strain gradient beam theory for thermo-mechanical dynamic characteristics of embedded smart shear deformable curved piezoelectric nanobeams made of porous electro-elastic functionally graded materials by using an analytical method. Electro-elastic properties of embedded curved porous FG nanobeam are assumed to be temperature-dependent and vary through the thickness direction of beam according to the power-law which is modified to approximate material properties for even distributions of porosities. It is perceived that during manufacturing of functionally graded materials (FGMs) porosities and micro-voids can be occurred inside the material. Since variation of pores along the thickness direction influences the mechanical and physical properties, so in this study thermo-mechanical vibration analysis of curve FG piezoelectric nanobeam by considering the effect of these imperfections is performed. Nonlocal strain gradient elasticity theory is utilized to consider the size effects in which the stress for not only the nonlocal stress field but also the strain gradients stress field. The governing equations and related boundary condition of embedded smart curved porous FG nanobeam subjected to thermal and electric field are derived via the energy method based on Timoshenko beam theory. An analytical Navier solution procedure is utilized to achieve the natural frequencies of porous FG curved piezoelectric nanobeam resting on Winkler and Pasternak foundation. The results for simpler states are confirmed with known data in the literature. The effects of various parameters such as nonlocality parameter, electric voltage, coefficient of porosity, elastic foundation parameters, thermal effect, gradient index, strain gradient, elastic opening angle and slenderness ratio on the natural frequency of embedded curved FG porous piezoelectric nanobeam are successfully discussed. It is concluded that these parameters play important roles on the dynamic behavior of porous FG curved nanobeam. Presented numerical results can serve as benchmarks for future analyses of curve FG nanobeam with porosity phases.

Nonlocal and Surface Effects on Vibration Behavior of Axially Loaded Flexoelectric Nanobeams Subjected to In-Plane Magnetic Field

In this research, vibration analysis of axially preloaded nonlocal flexoelectric nanobeams considering surface and magnetic field effects is performed. The main propose of present paper is to fill the gap in previous analyzes of flexoelectric nanobeams in which nonlocal and surface effects are ignored. Flexoelectricity in a nanobeam results in larger vibration frequencies than a typical piezoelectric nanobeam. Via Galerkin’s technique, the governing equations of the flexoelectric nanobeam are numerically solved. The vibration frequency of the nanobeam is validated with previous one in the literature. The illustrative results demonstrate the influences of the flexoelectricity, nonlocal stress field, surface energy, magnetic field, axial load and beam thickness on vibrational characteristics of a nanosize preloaded flexoelectric beam.

Dynamic response of porous functionally graded material nanobeams subjected to moving nanoparticle based on nonlocal strain gradient theory

Up to now, nonlocal strain gradient theory (NSGT) is broadly applied to examine free vibration, static bending and buckling of nanobeams. This theory captures nonlocal stress field effects together with the microstructure-dependent strain gradient effects. In this study, forced vibrations of NSGT nanobeams on elastic substrate subjected to moving loads are examined. The nanobeam is made of functionally graded material (FGM) with even and uneven porosity distributions inside the material structure. The graded material properties with porosities are described by a modified power-law model. Dynamic deflection of the nanobeam is obtained via Galerkin and inverse Laplace transform methods. The importance of nonlocal parameter, strain gradient parameter, moving load velocity, porosity volume fraction, type of porosity distribution and elastic foundation on forced vibration behavior of nanobeams are discussed.

Wave propagation analysis of rotating thermoelastically-actuated nanobeams based on nonlocal strain gradient theory

This paper is concerned with the wave propagation behavior of rotating functionally graded (FG) temperature-dependent nanoscale beams subjected to thermal loading based on nonlocal strain gradient stress field. Uniform, linear and nonlinear temperature distributions across the thickness are investigated. Thermo-elastic properties of FG beam change gradually according to the Mori–Tanaka distribution model in the spatial coordinate. The nanobeam is modeled via a higher-order shear deformable refined beam theory which has a trigonometric shear stress function. The governing equations are derived by Hamilton's principle as a function of axial force due to centrifugal stiffening and displacement. The solution of these equations is provided employing a Galerkin-based approach which has the potential to capture various boundary conditions. By applying an analytical solution and solving an eigenvalue problem, the dispersion relations of rotating FG nanobeam are obtained. Numerical results illustrate that various parameters including temperature change, angular velocity, nonlocality parameter, wave number and gradient index have significant effects on the wave dispersion characteristics of the nanobeam under study. The outcome of this study can provide beneficial information for the next-generation research and the exact design of nano-machines including nanoscale molecular bearings, nanogears, etc.

Longitudinal varying elastic foundation effects on vibration behavior of axially graded nanobeams via nonlocal strain gradient elasticity theory

ABSTRACTIn this research, vibration characteristics of axially functionally graded nanobeams resting on variable elastic foundation are investigated based on nonlocal strain gradient theory. This nonclassical nanobeam model contains a length scale parameter to explore the influence of strain gradients and also a nonlocal parameter to study the long-range interactions between the particles. The present model can degenerate into the classical models if the material length scale parameter and the nonlocal stress field parameter are both taken to be zero. Elastic foundation consists of two layers: a Winkler layer with variable stiffness and a Pasternak layer with constant stiffness. Linear, parabolic and sinusoidal variations of Winkler foundation in longitudinal direction are considered. Material properties are graded axially via a power-law distribution scheme. Hamilton's principle is employed to derive the governing equations that are solved applying a Galerkin-based solution for different boundary edges. Comparison study is also performed to verify the present formulation with those of previous papers. Results are presented to investigate the influences of the nonlocal and length scale parameters, various material compositions, elastic foundation parameters, type of foundation and various boundary conditions on the vibration frequencies of AFG nanobeams in detail.

A modified nonlocal couple stress-based beam model for vibration analysis of higher-order FG nanobeams

ABSTRACTIn the present paper, nonlocal couple stress theory is developed to investigate free vibration characteristics of functionally graded (FG) nanobeams considering exact position of neutral axis. The theory introduces two parameters based on nonlocal elasticity theory and modified couple stress theory to capture the size effects much accurately. Therefore, a nonlocal stress field parameter and a material length scale parameter are used to involve both stiffness-softening and stiffness-hardening effects on responses of FG nanobeams. The FG nanobeam is modeled via a higher-order refined beam theory in which shear deformation effect is verified needless of shear correction factor. A power-law distribution is used to describe the graded material properties. The governing equations and the related boundary conditions are derived by Hamilton's principle and they are solved applying Galerkin's method, which satisfies various boundary conditions. A comparison study is performed to verify the present formulation with the provided data in the literature and a good agreement is observed. The parametric study covered in this paper includes several parameters, such as nonlocal and length scale parameters, power-law exponent, slenderness ratio, shear deformation, and various boundary conditions on natural frequencies of FG nanobeams in detail.

A nonlocal strain gradient hyperbolic shear deformable shell model for radial postbuckling analysis of functionally graded multilayer GPLRC nanoshells

The present study addresses the size dependency in nonlinear instability of functionally graded multilayer graphene platelet-reinforced composite (GPLRC) nanoshells under hydrostatic pressure including jointly the nonlocal elastic and strain gradients stress fields. For this objective, the new unconventional continuum theory namely as nonlocal strain gradient theory of elasticity is utilized within the framework of a refined hyperbolic shear deformation shell theory. Via stacking up a number of individual layers, the graphene platelet (GPL) nanofillers are distributed uniformly and three different functionally graded patterns based upon a layerwise change of the GPL weight fraction through the shell thickness direction. The effective material properties corresponding to uniform (U-GPLRC) and X-GPLRC, O-GPLRC, A-GPLRC functionally graded patterns of dispersion are extracted by Halpin-Tsai micromechanical scheme. By employing jointly the boundary layer theory of shell buckling and a two-stepped perturbation technique, explicit analytical expressions are achieved for nonlocal strain gradient stability curves of functionally graded multilayer GPLRC nanoshells. It is indicated that by increasing the value of GPL weight fraction for the U-GPLRC and O-GPLRC nanoshells, the significance of the both nonlocal and strain gradient size dependencies reduces, while for the X-GPLRC and A-GPLRC nanoshells, it increases.

Cylindrical functionally graded shell model based on the first order shear deformation nonlocal strain gradient elasticity theory

In this article, a cylindrical functionally graded shell model is developed in the framework of nonlocal strain gradient theory for the first time. For this purpose, the modeled cylindrical FG nanoshell, its equations of motion and corresponding boundary conditions are derived by Hamilton’s principle and the first-order shear deformation theory. Generalized differential quadrature method is applied to discretize the equations of motion. The results of the proposed model are compared with those of the Eringen’s nonlocal, strain gradient, modified couple stress and classical theories. The conclusion of this comparison is that the nonlocal strain gradient theory combines advantages of nonlocal and strain gradient theories by applying both material length scale parameter and a nonlocal parameter in the model to consider the significance of strain gradient stress field and nonlocal elastic stress field, respectively. Furthermore, the effects of the material length scale, nonlocal parameter, FG power index, circumferential wave numbers and length of shell on vibrational behavior of the nonlocal strain gradient FG nanoshell for simply supported and clamped–clamped boundary conditions are investigated.

Buckling analysis of nonlocal strain gradient axially functionally graded nanobeams resting on variable elastic medium

This paper provides the first examination of buckling behavior of axially functionally graded nanobeams. A nonlocal strain gradient theory consisting of two scale parameter is employed for modeling of size-dependent behavior axially functionally graded nanobeam much accurately. This theory takes into account both nonlocal stress field and strain gradient effects on the response of nanostructures. A power-law model is used to describe the distribution of material properties along the axial direction. The axially functionally graded nanobeam is in contact with a non-uniform elastic medium which consists of a Winkler layer with variable stiffness and also a Pasternak layer with constant stiffness. Linear, parabolic and sinusoidal variations of Winkler foundation in longitudinal direction are considered. A Galerkin-based solution technique is implemented to solve the governing equation obtained from Hamilton’s principle. Buckling loads of functionally graded nanobeam are verified with those of previous papers. It is shown that buckling loads of axially functionally graded nanobeams are significantly influenced by power-law index, nonlocal parameter, length scale parameter, type of elastic foundation and boundary conditions.

Vibration analysis of piezoelectrically actuated curved nanosize FG beams via a nonlocal strain-electric field gradient theory

ABSTRACTIn this article, nonlocal free vibration analysis of curved functionally graded piezoelectric (FGP) nanobeams is conducted using a Navier-type solution method. The model contains a nonlocal stress field parameter and also a nonlocal strain-electric field gradient parameter to capture the size effects. Inclusion of these nonlocal parameters introduces both stiffness-softening and stiffness-hardening effects in the analysis of curved nanobeams. Nonlocal governing equations of curved FGP nanobeam are obtained from Hamilton's principle based on the Euler–Bernoulli beam model. The results are validated with those of curved FG nanobeams available in the literature. Finally, the influences of electric voltage, length scale parameter, nonlocal parameter, opening angle, material composition, and slenderness ratio on vibrational characteristics of nanosize curved FG piezoelectric beams are explored. These results may be useful in accurate analysis and design of smart nanostructures constructed from piezoelectric materials.

Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory

A size-dependent inhomogeneous beam model, which accounts for the through-length power-law variation of a two-constituent axially functionally graded (FG) material, is deduced in the framework of the nonlocal strain gradient theory and the Euler–Bernoulli beam theory. By employing the Hamilton principle, the equations of motion and boundary conditions for size-dependent axially FG beams are deduced. A material length scale parameter and a nonlocal parameter are introduced in the axially FG beam model to consider the significance of strain gradient stress field and nonlocal elastic stress field, respectively. The bending, buckling and vibration problems of axially FG beams are solved by a generalized differential quadrature method. The influences of power-law variation and size-dependent parameters on the bending, buckling and vibration behaviors of axially FG beams are investigated. The mechanical behaviors can be affected by the through-length grading of the FG material and therefore may be controlled by choosing appropriate values of the power-law index. When considering concentrated and uniformly distributed loads, the maximum deflection decreases with increasing length scale parameter. The axially FG beam may exert a stiffness-softening effect or a stiffness-hardening effect on the critical buckling force and the natural frequencies depending on the values of the two size-dependent parameters.

Vibration analysis of viscoelastic inhomogeneous nanobeams incorporating surface and thermal effects

This article deals with the free vibration investigation of nonlocal strain gradient-based viscoelastic functionally graded (FG) nanobeams on viscoelastic medium considering surface stress effects. Nonlocal strain gradient theory possesses a nonlocal stress field parameter and a length scale parameter for more accurate prediction of mechanical behavior of nanostructures. Surface energy effect is incorporate to the nonlocal strain gradient theory employing Gurtin–Murdoch elasticity theory. Thermo-elastic material properties of nanobeam are graded in thickness direction using power-law distribution. Hamilton’s principal is utilized to obtain the governing equations of FG nanobeam embedded in viscoelastic medium. The effects of surface stress, length scale parameter, nonlocal parameter, viscoelastic medium, internal damping constant, thermal loading, power-law index and boundary conditions on vibration frequencies of viscoelastic FGM nanobeams are discussed in detail.

Through-the-length temperature distribution effects on thermal vibration analysis of nonlocal strain-gradient axially graded nanobeams subjected to nonuniform magnetic field

ABSTRACTThermomechanical vibration analysis of axially functionally graded (AFG) nanobeams under nonuniform longitudinal magnetic field is investigated based on nonlocal strain-gradient theory. This theory contains two scale parameters for modeling of size-dependent behavior of AFG nanobeam accurately. This theory takes into account both nonlocal stress field and strain-gradient effects on the response of nanostructures. The nanobeam is subjected to uniform and linear through-the-length temperature distributions. A power-law model is used to describe the distribution of temperature-dependent material properties along the axial direction. A Galerkin-based solution technique is implemented to solve the governing equation obtained from Hamilton’s principle. Natural frequencies of functionally graded nanobeam are verified with those of previous articles. It is shown that vibration frequencies of AFG nanobeams are significantly influenced by temperature rise, power-law index, nonlocal parameter, length-scale parameter, magnetic field intensity, and boundary conditions.

Vibration analysis of viscoelastic inhomogeneous nanobeams resting on a viscoelastic foundation based on nonlocal strain gradient theory incorporating surface and thermal effects

In this paper, surface and thermal effects on the vibration characteristics of viscoelastic functionally graded (FG) nanobeams embedded in a viscoelastic foundation are investigated based on nonlocal strain gradient elasticity theory and Euler–Bernoulli beam model. The present model contains two parameters to capture the size effects in which the stress is considered for not only the nonlocal stress field but also the strain gradients stress field. Gurtin–Murdoch elasticity theory is incorporated into the nonlocal strain gradient theory to develop a nonclassical beam model including the surface effects. The viscoelastic foundation consists of a Winkler–Pasternak layer together with a viscous layer of infinite parallel dashpots. A power-law model is adopted to describe a continuous variation of material properties of the FG nanobeam. The governing equations of nonlocal viscoelastic FG nanobeams are obtained using Hamilton’s principle and solved implementing an analytical solution for simply supported and clamped–clamped boundary conditions. The results are validated with those available in the literature. The effects of linear, shear and viscous layers of foundation, structural damping coefficient, surface elasticity, nonlocal parameter, length scale parameter, temperature change, power-law exponent, and slenderness ratio on the frequencies of viscoelastic FG nanobeams are examined.

A first passage based model for probabilistic fracture of polycrystalline silicon MEMS structures

Experiments have shown that the failure loads of Microelectromechanical Systems (MEMS) devices usually exhibit a considerable level of variability, which is believed to be caused by the random material strength and the geometry-induced random stress field. Understanding the strength statistics of MEMS devices is of paramount importance for the device design guarding against a tolerable failure risk. In this study, we develop a continuum-based probabilistic model for polycrystalline silicon (poly-Si) MEMS structures within the framework of first passage analysis. The failure of poly-Si MEMS structures is considered to be triggered by fracture initiation from the sidewalls governed by a nonlocal failure criterion. The model takes into account an autocorrelated random field of material tensile strength. The nonlocal random stress field is obtained by stochastic finite element simulations based on the information of the uncertainties of the sidewall geometry. The model is formulated within the contexts of both stationary and non-stationary stochastic processes for MEMS structures of various geometries and under different loading configurations. It is shown that the model agrees well with the experimentally measured strength distributions of uniaxial tensile poly-Si MEMS specimens of different gauge lengths. The model is further used to predict the strength distribution of poly-Si MEMS beams under three-point bending, and the result is compared with the Monte Carlo simulation. The present model predicts strong size effects on both the strength distribution and the mean structural strength. It is shown that the mean size effect curve consists of three power-law asymptotes in the small, intermediate, and large-size regimes. By matching these three asymptotes, an approximate size effect equation is proposed. The present model is shown to be a generalization of the classical weakest-link statistical model, and it provides a physical interpretation of the material length scale used in the weakest-link model.

Thermal effects on wave propagation characteristics of rotating strain gradient temperature-dependent functionally graded nanoscale beams

ABSTRACTIn the present article, wave dispersion behavior of a temperature-dependent functionally graded (FG) nanobeam undergoing rotation subjected to thermal loading is investigated according to nonlocal strain gradient theory, in which the stress enumerates for both nonlocal stress field and the strain gradient stress field. Mori–Tanaka distribution model is considered to express the gradual variation of material properties across the thickness. The governing equations are derived as a function of axial force due to centrifugal stiffening and displacements by applying Hamilton’s principle according to Euler–Bernoulli beam theory. By applying an analytical solution, the dispersion relations of rotating FG nanobeam are obtained by solving an eigenvalue problem. Obviously, numerical results indicate that various parameters such as angular velocity, gradient index, temperature change, wave number, and nonlocality parameter have significant influences on the wave characteristics of rotating FG nanobeams. Hence, the results of this research can provide useful information for the next generation studies and accurate design of nanomachines including nanoscale molecular bearings and nanogears, etc.

Thermal environment effects on wave dispersion behavior of inhomogeneous strain gradient nanobeams based on higher order refined beam theory

ABSTRACTIn this article, an analytical model for the wave propagation analysis of inhomogeneous functionally graded (FG) nanobeam in thermal environment is developed based on nonlocal strain gradient theory, in which the stress accounts for not only the nonlocal elastic stress field but also the strain gradients stress field. The nanobeam is modeled through a higher order shear deformable refined beam theory which has a trigonometric shear stress function. The temperature field supposed to have a nonlinear distribution across the nanobeam thickness. Temperature-dependent material properties of nanobeams are spatially graded based on Mori–Tanaka model. The governing equations of the temperature-dependent functionally graded (FG) nanobeam are derived using the Hamilton’s principle. Numerical examples show that the characteristics of the wave propagation of FG nanobeam are influenced by various parameters such as nonlocality parameter, length scale parameter, gradient index, and temperature changes.

A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams

This paper investigates buckling characteristics of a curved functionally graded (FG) nanobeam based on nonlocal strain gradient elasticity theory accounting the stress for not only the nonlocal stress field but also the strain gradients stress field. The modeling of nanobeam is carried out via a higher order refined beam theory which captures shear deformation influences needless of any shear correction factor. Power-law model is adopted to describe continuous variation of material properties of curved FG nanobeam. The governing equations of nonlocal strain gradient curved FG nanobeam in the framework of refined hyperbolic beam model are obtained using Hamilton’s principle and solved implementing an analytical solution for simply-supported and clamped boundary conditions. To validate the present model, the results are compared with those of straight FG nanobeams by extending the radius of nanobeam to infinity. The effects of nonlocal parameter, length scale parameter, power-law exponent, boundary conditions and slenderness ratio on the buckling response of curved FG nanobeams are investigated.

Size-dependent dynamic modeling of inhomogeneous curved nanobeams embedded in elastic medium based on nonlocal strain gradient theory

In this paper, size-dependent free vibration analysis of curved functionally graded nanobeams embedded in Winkler–Pasternak elastic medium is carried out via an analytical solution method. Three kinds of boundary condition namely, simply supported-simply supported, simply supported-clamped and clamped-clamped are investigated. Material properties of curved functionally graded beam change in thickness direction according to the Mori–Tanaka model. Nonlocal strain gradient elasticity theory is adopted to capture the size effects in which the stress is considered for not only the nonlocal stress field but also the strain gradients stress field. Nonlocal governing equations of curved functionally graded nanobeam are obtained from Hamilton’s principle based on Euler–Bernoulli beam model. Finally, the influences of length scale parameter, nonlocal parameter, opening angle, elastic medium, material composition, slenderness ratio and boundary conditions on the vibrational characteristics of nanosize curved functionally graded beams are explored.