Strain Gradient Beam Research Articles

Size-Dependent Forced Vibration Analysis of Three Nonlocal Strain Gradient Beam Models with Surface Effects Subjected to Moving Harmonic Loads

The forced vibration behaviors are examined for nonlocal strain gradient nanobeams with surface effects subjected to a moving harmonic load travelling with a constant velocity in terms of three beam models namely, the Euler-Bernoulli, Timoshenko and modified Timoshenko beam models. The modification for nonlocal strain gradient Timoshenko nanobeams is exerted to the constitutive equations by exclusion of the nonlocality in the shear constitutive relation. Some analytical closed-form solutions for three nonlocal strain gradient beam models with simply supported boundary conditions are derived by using the Galerkin discretization method in conjunction with the Laplace transform method. The effects of the three beam models, the nonlocal and material length scale parameters, the velocity and excitation frequency of the moving harmonic load on the dynamic behaviors of nanobeams are discussed in some detail. Specifically, the critical velocities are examined in some detail. Numerical results have shown that the aforementioned parameters are very important factors for determining the dynamic behavior of the nanobeams accurately.

Vibration of nonlocal strain gradient beams incorporating Poisson's ratio and thickness effects

Poisson's ratio is an extremely important physical parameter of materials, which can be defined as the ratio of transverse strain to longitudinal strain. Due to the fact that the Poisson's ratio for common structural materials is within a very narrow domain (i.e., from 0.25 to 0.35), the Poisson's ratio effect on mechanical behaviors of nanobeams is few investigated. However, the Poisson's ratio for isotropic solids is from −1 to 0.5, and the structures with special Poisson's ratio, especially the negative Poisson's ratio, have attracted wide attention. Thus, it is necessary to study the Poisson's ratio effect on mechanical behaviors of nanobeams. Free vibration of nanobeams was performed by many works based on nonlocal strain gradient theory (NSGT), but most of them did not consider the size-dependent effect in thickness direction. In this study, the effects of Poisson's ratio and thickness are examined for the vibration behaviors of nanobeams via NSGT. Based on NSGT with the thickness effect, the analytical solution for the bending frequencies of hinged-hinged nanobeams are derived. When neglecting the thickness effect in NSGT, the stiffness-hardening effect is underestimated, especially for thin beams where the size-dependent effect of loss of neighboring atoms on the beam's surface is significant. By numerical results and the molecular dynamics simulation, it is found that the stiffness-hardening effect in [110] copper nanowire attenuates with an increased vibrational mode number. Moreover, the Poisson's ratio effect in vibration frequency of reflexyne and flexyne nanobeams is different. The Poisson's ratio effect in flexyne and reflexyne nanobeams with the thickness effect is extraordinarily remarkable.

Size-dependent nonlinear secondary resonance of micro-/nano-beams made of nano-porous biomaterials including truncated cube cells

Porous biomaterials have been utilized in cellular structures in order to mimic the function of bone as a branch of tissue engineering approach. With the aid of nanoporous biomaterials in which the pore size is at nanoscale, the capability of biological molecular isolation becomes more efficient. In the present study, firstly the mechanical properties of nanoporous biomaterials are estimated on the basis of a truncated cube cell model including a refined hyperbolic shear deformation for the associated lattice structure. After that, based upon a nonlocal strain gradient beam model, the size-dependent nonlinear secondary resonance of micro/nano-beams made of the nanoporous biomaterial is predicted corresponding to the both of subharmonic and superharmonic excitations. The non-classical governing differential equation of motion is constructed via Hamilton's principle. By employing the Galerkin technique together with the multiple time-scales method, the nonlocal strain gradient frequency-response and amplitude-response of the nonlinear oscillation of micro/nano-beams made of a nanoporous biomaterial under hard excitation are achieved. It is shown that in the superharmonic case, increasing the pore size leads to enhance the nonlinear hardening spring-type behavior of jump phenomenon and the height of limit point bifurcations. In the subharmonic case, higher pore size causes to increase the gap between two branches associated with the high-frequency and low-frequency solutions.

Coupling effect of thickness and shear deformation on size-dependent bending of micro/nano-scale porous beams

• A unified nonlocal strain gradient beam model with the thickness effect is developed. • Analytical solutions for the bending of hinged-hinged beams are obtained. • Size effect can be not only length-dependent but also thickness-dependent. • The generalized Young’s modulus depends on applied load types. • The coupling of the shear and thickness effects needs be drawn attention. A unified nonlocal strain gradient beam model with the thickness effect is developed to investigate the static bending behavior of micro/nano-scale porous beams. Size-dependent governing equations and corresponding analytical solutions for the bending of hinged-hinged beams are obtained by employing minimum total potential energy principle, the Navier solution method as well as the variational-consistent boundary conditions. For nonlocal strain gradient theory (NSGT) with thickness effect, virtual strain energy function of shear beams can contain additional nonlocal shear stress and high-order nonlocal shear stress related to the thickness direction in comparison with that of Euler–Bernoulli beam, so the coupling of the shear and thickness effects should be drawn huge attention. By means of detailed numerical analysis, it is found that, the stiffness-hardening effect is underestimated in NSGT without the thickness effect, and the stiffness-hardening and stiffness-softening effects of NSGT with the thickness effect can be not only length-dependent but also thickness-dependent. Interestingly, the generalized Young’s modulus depends on half-wave number, which means that the generalized Young’s modulus may be different due to applied load types. In the context of NSGT with the thickness effect, the deflection of Euler–Bernoulli beam predicted is smaller than that of shear beam, especially for thick beams. Furthermore, porosities distributed in the top or bottom of beams can possess a greater influence on the decrease of overall stiffness of beam than those distributed in the vicinity of the middle plane of beams.

Stress gradient, strain gradient and inertia gradient beam theories for the simulation of flexural wave dispersion in carbon nanotubes

Flexural wave propagation in carbon nanotubes (CNTs) can be described through higher-order elasticity theories so as to capture the dispersive behavior induced by the inherent nanoscale heterogeneity. Motivated by experimental dispersion characteristics of metal nano-structured crystals, a new three-length-scale gradient formulation has been recently developed by the authors. In addition to the Laplacian of the strain, this model incorporates two higher-order inertia gradients for an improved dispersion behavior. A discrete medium with lumped masses at multiple scales of observation and combination of lumped mass and distributed mass at the macro-scale is introduced here to provide a micro-mechanical background to the proposed three-length-scale gradient model. The next aim of this paper is to assess the ability of this model to simulate flexural wave dispersion occurring in CNTs. We employ gradient-enriched Euler-Bernoulli and Timoshenko beam theories incorporating either stress gradients, or a combination of both strain gradients and inertia gradients – the latter leading to novel gradient-enriched beam theories. It is demonstrated that the proposed three-length-scale gradient elasticity formulation is able to capture the wave dispersion characteristics of armchair single-walled (5,5) and (10,10) CNTs arising from Molecular Dynamics simulations with high accuracy for a wide range of wave numbers. Advantages over alternative formulations of higher-order beam theories with stress gradients or combined strain-inertia gradient enrichments are discussed for comparative purposes.

Reissner stationary variational principle for nonlocal strain gradient theory of elasticity

Abstract The general form of Reissner stationary variational principle is established in the framework of the nonlocal strain gradient theory of elasticity. Including two size-dependent characteristic parameters, the nonlocal strain gradient elasticity theory can demonstrate the significance of the strain gradient as well as the nonlocal elastic stress field. Based on the Reissner functional, the governing differential and boundary conditions of dynamic equilibrium and differential constitutive equations of the classical and first-order nonlocal stress tensor are derived in the most general form. Additionally, the boundary congruence conditions are formulated and discussed for the nonlocal strain gradient theory. To exhibit the application value of Reissner variational principle, it is employed to examine the nonlinear vibrations of size-dependent Bernoulli-Euler and Timoshenko beams. In the case of immovable boundary conditions, employing the weighted residual Galerkin method, the homotopy analysis method is also utilized to determine the closed form analytical solutions of the geometrically nonlinear vibration equations. Consequently, the analytical expressions for the nonlinear natural frequencies of Bernoulli-Euler and Timoshenko nonlocal strain gradient beams are derived.

Analytical solutions for the thermal vibration of strain gradient beams with elastic boundary conditions

A strain gradient Euler beam described by a sixth-order differential equation is used to investigate the thermal vibrations of beams made of strain gradient elastic materials. The sixth-order differential equation of motion and elastic boundary conditions are determined simultaneously by a variation formulation based on Hamilton’s principle. Analytical solutions for the free vibration of the elastic constraint strain gradient beams subjected to axial thermal stress are obtained. The effects of the thermal stress, nonlocal effect parameter, and boundary spring stiffness on the vibration behaviors of the strain gradient beams are investigated. The results show that the natural frequencies obtained by the strain gradient Euler beam model with the thermal stress decrease while the temperature is rising. The thermal effects are sensitive to the boundary spring stiffness at a certain stiffness range. In addition, numerical results also show the importance of the nonlocal effect parameter on the vibration of the strain gradient beams.

Nonlinear primary resonance of micro/nano-beams made of nanoporous biomaterials incorporating nonlocality and strain gradient size dependency

Abstract A wide range of biological applications such as drug delivery, biosensors and hemodialysis can be provided by nanoporous biomaterials due to their uniform pore size as well as considerable pore density. In the current study, the size dependency in the nonlinear primary resonance of micro/nano-beams made of nanoporous biomaterials is anticipated. To accomplish this end, a refined truncated cube is introduced to model the lattice structure of nanoporous biomaterial. Accordingly, analytical expressions for the mechanical properties of material are derived as functions of pore size. After that, based upon a nonlocal strain gradient beam model, the size-dependent nonlinear Duffing type equation of motion is constructed. The Galerkin technique together with the multiple time-scales method is employed to obtain the nonlocal strain gradient frequency-response and amplitude-response related to the nonlinear primary resonance of a micro/nano-beam made of the nanoporous biomaterial with different pore sizes. It is indicated that the nonlocality causes to decrease the response amplitudes associated with the both bifurcation points of the jump phenomenon, while the strain gradient size dependency causes to increase them. Also, it is found that increasing the pore size leads to enhance the nonlinearity, so the maximum deflection of response occurs at higher excitation frequency.

Forced vibration response in nanocomposite cylindrical shells - Based on strain gradient beam theory

In this paper, forced vibration of micro cylindrical shell reinforced by functionally graded carbon nanotubes (FG-CNTs) is presented. The structure is subjected to transverse harmonic load and modeled by beam model. The size effects are considered based on strain gradient theory containing three small scale parameters. The mixture rule is used for obtaining the effective material properties of the structure. Based on sinusoidal shear deformation theory of beam, energy method and Hamilton\'s principle, the motion equations are derived. Applying differential quadrature method (DQM) and Newmark method, the frequency curves of the structure are plotted. The effect of different parameters including, CNTs volume percent and distribution type, boundary conditions, size effect and length to thickness ratio on the frequency curves of the structure is studied. Numerical results indicate that the dynamic deflection of the FGX-CNT-reinforced cylindrical is lower with respect to other type of CNT distribution.

The effect of thickness on the mechanics of nanobeams

Nonlocal strain gradient models are being used more and more extensively in examining the size-dependent effects on the statical and dynamical behaviors of micro/nano-structures. However, for the fake of simplification, the size-dependent effects are often assumed to be neglected in the thickness direction of beams and plates. This simplification is originally aiming at reducing the complexity of the studied beam and plate problems while retaining all the most important mechanical features. Based on the nonlocal strain gradient model without the thickness effect, the size-dependent effect may reveal stiffness-softening or stiffness-hardening effect, which only depends on the values of strain-gradient and stress-gradient parameters but is independent on the geometric feature. In this paper, a nonlocal strain gradient beam model incorporating the thickness effect is developed for the size-dependent buckling analysis of nanobeams, and closed-form solutions are derived for post-buckling configuration and critical buckling force. When incorporating the size-dependent effect of thickness, it is found that the stiffness-softening and stiffness-hardening effects depend not only on the ratio of stress-gradient parameter to strain-gradient parameter, but also on the geometric feature (slenderness ratio). Interestingly, the stiffness-softening effect can be only found for the longitudinal dispersion relation of Silicon, while the stiffness-hardening effect may be observed for the buckling behaviors of nanobeams made of Silicon due to the size-dependent effect of thickness.

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 strain gradient beam model for postbuckling and associated vibrational response of lipid supramolecular protein micro/nano-tubules

As a supramolecular construction, lipid protein micro/nano-tubules can be utilized in a variety of sustained biological delivery system. The high slenderness ratio of lipid tubules makes their hierarchical assembly into a desired architecture difficult. Therefore, an accurate prediction of mechanical behavior of lipid tubular is essential. The objective of this study is to capture size dependency in the postbuckling and vibrational response of the postbuckled lipid micro/nano-tubules more comprehensively. To this purpose, the nonlocal strain gradient elasticity theory is incorporated to the third-order shear deformation beam theory to develop an unconventional beam model. Hamilton's principle is put to use to establish the size-dependent governing differential equations of motion. After that, an improved perturbation technique in conjunction with Galerkin method is employed to obtain the nonlocal strain gradient load-frequency response and postbuckling stability curves of lipid micro/nano-tubules. It is revealed that by taking the nonlocal size effect into consideration, the influence of the type (geometrical parameters) of an axially compressed lipid micro/nano-tubule on its natural frequency in order decreases and increases within the prebuckling and postbuckling regimes. While the strain gradient size dependency plays an opposite role which causes that the influence of the type of lipid micro/nano-tubule on its natural frequency corresponding to the prebuckling and postbuckling domains increases and decreases, respectively.

Dynamic transverse vibration characteristics of nonuniform nonlocal strain gradient beams using the generalized differential quadrature method

In this study, vibration response of small-scale tapered beams is investigated. Size-dependent effects are modeled in the framework of the nonlocal strain gradient theory and three different types of linearly cross-section variation are proposed by having width variation, thickness variation and a combination of them both. The small-scale beam is formulated using Euler-Bernoulli beam theory, nonlocal strain gradient theory and Hamilton’s principle. Equations of motion for all three types of nonuniformity are solved using the generalized differential quadrature method (GDQM). Results are presented and compared to those achieved for simplified models and effects of having different slenderness ratio are presented. Moreover, a comprehensive parametric study is proposed and the effects of varying nonuniformity terms, nonlocal and strain gradient parameters are precisely studied. Accordingly, with the vast application of tapered small-scale beams in many devices, especially scanning tunneling microscopes (STM) and atomic force microscopes (AFM), this study could be a step forward in understanding, predicting and controlling such kind of behaviors.

Bending, buckling and vibration of small-scale tapered beams

In this paper, a comprehensive study on mechanical behavior of non-uniform small scale beams in the framework of nonlocal strain gradient theory is presented. The governing equations and boundary conditions have been developed using Hamilton's principle and solved with the aid of finite element method for introducing the bending, buckling and free vibration response of nano-beams. The current formulation could be used for all types of non-uniformities by varying the width and thickness of nano-beams along the length. The accuracy of the current model and formulation is verified by comparing the results with previous literature and those obtained by analytically solving simplified problems. In order to understand the influence of having non-uniform cross section on static and dynamic behavior of small scale beams, parametric study is presented and the effects of different parameters such as non-uniformity, non-local and strain gradient terms on natural frequency, buckling load and deformation are observed and discussed. It is seen that having non-uniform cross section in nonlocal strain gradient beams could lead to significant changes in mechanical behavior of such structures.

Nonlocal strain gradient beam model for nonlinear vibration of prebuckled and postbuckled multilayer functionally graded GPLRC nanobeams

In the current study, a nonlocal strain gradient beam model with third-order distribution of shear deformation is established to explore the nonlinear vibration of axially-loaded multilayer functionally graded graphene platelet-reinforced composite (GPLRC) nanobeams in both of the prebuckling and postbuckling domains. The dispersion of graphene platelet (GPL) nanofillers changes layerwise based upon different functionally graded patterns while it remains constant within each individual layer. The effective mechanical properties of multilayer functionally graded GPLRC nanobeams are estimated using Halpin-Tsai model of micromechanics. Hamilton's principle is utilized to construct the size-dependent differential equations of motion. Subsequently, an improved perturbation technique in conjunction with the Galerkin method is employed to present explicit analytical expression for nonlocal strain gradient nonlinear frequency in terms of applied axial load. It is observed that at the critical buckling point, the significance of the nonlocality and strain gradient size dependency on the nonlinear frequency remains constant for all values of maximum deflection. However, within the prebuckling and postbuckling regimes, by increasing the maximum deflection of the axially-loaded multilayer GPLRC nanobeam, both types of size effects on the nonlinear frequency reduce. Also, it is seen that similar to the type of GPL dispersion pattern, the value of GPL weight fraction has also no influence on the significance of size dependencies in the nonlinear frequency of axially-loaded multilayer functionally graded GPLRC nanobeams.

Buckling analysis of nonuniform nonlocal strain gradient beams using generalized differential quadrature method

Recently, it was shown that the length scales presented in nonlocal elasticity and strain gradient theory each describe a different physical and material properties of structures at small scales. Accordingly, in this article, by using the new accurate nonlocal strain gradient theory, buckling behavior of nonuniform small-scale beam is investigated. Nanobeam is assumed with variable cross section through the length by exponentially varying the width. According to the size effects, higher order strain deformations and nonlocal effects are modeled on the Euler-Bernoulli beam. Governing equation of motion is presented using Hamilton's principle for nonuniform nonlocal strain gradient beams and solved using generalized differential quadrature method for higher order differential equations. Accuracy of the current methodology is discussed by increasing the number of sampling points and merging to unique solutions. Moreover, in order comprehend the nonuniformity effects on nonlocal strain gradient beams, parametric study is done and presented for different variables. It is shown that nonuniformity could have a significant efficacy on critical buckling loads depending on the ratio between nonlocal and strain gradient parameters. This study is a step forward in better understanding the behavior of nonuniform small scale beams to be used in different nanoscale structures.

Bending and buckling of nonlocal strain gradient elastic beams

Featured by the two material length parameters in the nonlocal strain gradient theory, it is still unknown that what are the boundary conditions of nonlocal strain gradient beams, since the equations of motion and boundary conditions of these beam models appear in the same form as those of the classical ones. Based on the weighted residual approaches, this paper provides the boundary value problems of Euler–Bernoulli beams within the framework of the nonlocal strain gradient theory in conjunction with the von Kármán nonlinear geometric relation. The closed-form solutions for bending and buckling loads of nonlocal strain gradient beams are obtained. Numerical results show that the higher-order boundary conditions have no effect on the static bending deflection of beams for the cases studied. However, the higher-order boundary conditions and the material length parameters have a significant effect on the buckling loads. Finally, when the two material length parameters are the same, the buckling loads can not always reduce to the classical solutions, the findings of which violate our expectations. The results provided in this work are expected to be helpful for the applications of this theory to the analysis of engineering structures.

Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material

Size-dependent nonlinear Euler-Bernoulli and Timoshenko beam models, which account for the through-thickness power-law variation of two-constituent functionally graded (FG) materials, are derived to investigate the nonlinear bending and free vibration behaviors in the framework of the nonlocal strain gradient theory. The nonlinearity due to the stretching effect of the mid-plane of the FG beam is the source of nonlinearity of the considered bending and free vibration problems. The size-dependent equations of motion and boundary conditions are derived by employing the Hamilton’s principle. The beam models contain material length scale and nonlocal parameters to consider the effects of both inter-atomic long-range force and microstructure deformation mechanism. In the case of hinged-hinged boundary conditions, the analytical solutions for the nonlinear bending deflection and free vibration frequencies of nonlocal strain gradient Euler-Bernoulli and Timoshenko beams are deduced. The influences of the through-thickness power-law variation of a two-constituent FG material and size-dependent parameters on nonlinear bending deflection and free vibration frequencies are investigated. Due to the intrinsic stiffening effect brought by the stretching effect of the mid-plane of the beam, the nonlinear bending deflections are smaller than their linear counterparts under the action of the same force, while the nonlinear vibration frequencies are higher than their linear counterparts for the same amplitude of the nonlinear oscillator. The nonlinear bending deflections and free vibration frequencies can be affected significantly by the through-thickness grading of FG materials in the beam. When the nonlocal parameter is smaller than the material characteristic parameter, the nonlinear FG beam reveals a stiffness-hardening effect. When the material characteristic parameter is smaller than the nonlocal parameter, the FG beam reveals a stiffness-softening effect.

Free vibration analysis of nonlocal strain gradient beams made of functionally graded material

A size-dependent Timoshenko beam model, which accounts for through-thickness power-law variation of a two-constituent functionally graded (FG) material, is derived in the framework of the nonlocal strain gradient theory. The equations of motion and boundary conditions are deduced by employing the Hamilton principle. The model contains a material length scale parameter introduced to consider the significance of strain gradient stress field and a nonlocal parameter introduced to consider the significance of nonlocal elastic stress field. The influence of through-thickness power-law variation and size-dependent parameters on vibration is investigated. It is found that through-thickness grading of the FG material in the beam has a great effect on the natural frequencies and therefore can be used to control the natural frequencies. The vibration frequencies can generally increase with the increasing material length scale parameter or the decreasing nonlocal parameter. When the material characteristic parameter is smaller than the nonlocal parameter, the FG beam exerts a stiffness-softening effect. When the material characteristic parameter is larger than the nonlocal parameter, the FG beam exerts a stiffness-hardening effect.

Closed-form frequency solutions for simplified strain gradient beams with higher-order inertia

This paper develops an Euler−Bernoulli beam model within the context of a simplified strain gradient theory with higher-order inertia. In contrast to the classical beam models, the proposed gradient beam models can capture the size effects by introducing not only the internal length l2 related to strain gradient but also the internal length l1 related to velocity gradient. The governing equation of motion and boundary conditions are derived by using the variational principles. The closed-form solutions for free vibrations of beams with three typical boundary conditions are obtained. Numerical results show that the choices of the higher-order boundary conditions have a minor effect on the natural frequencies of beams. In addition, the inclusion of the strain gradient parameter l2 increases the effective stiffness of beams and hence it increases the natural frequencies of beams; whereas the inclusion of the velocity gradient parameter l1 acts as an equivalent compression force in the governing equation and therefore it leads to the decrease of the natural frequencies of beams. Moreover, the significant Poisson effect on the natural frequencies of beams is observed when the thickness of the beam is comparable to the strain gradient parameter l2. The closed-form solutions for natural frequencies of beams presented in this work may serve as benchmark results for other numerical methods.