On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory： equilibrium, governing equation and static deflection

This paper has successfully addressed three critical but overlooked issues in nonlocal elastic stress field theory for nanobeams： （i） why does the presence of increasing nonlocal effects induce reduced nanostructural stiffness in many, but not consistently for all, cases of study, i.e., increasing static deflection, decreasing natural frequency and decreasing buckling load, although physical intuition according to the nonlocal elasticity field theory first established by Eringen tells otherwise？ （ii） the intriguing conclusion that nanoscale effects are missing in the solutions in many exemplary cases of study, e.g., bending deflection of a cantilever nanobeam with a point load at its tip; and （iii） the non-existence of additional higher-order boundary conditions for a higher-order governing differential equation. Applying the nonlocal elasticity field theory in nanomechanics and an exact variational principal approach, we derive the new equilibrium conditions, do- main governing differential equation and boundary conditions for bending of nanobeams. These equations and conditions involve essential higher-order differential terms which are opposite in sign with respect to the previously studies in the statics and dynamics of nonlocal nano-structures. The difference in higher-order terms results in reverse trends of nanoscale effects with respect to the conclusion of this paper. Effectively, this paper reports new equilibrium conditions, governing differential equation and boundary condi- tions and the true basic static responses for bending of nanobeams. It is also concluded that the widely accepted equilibrium conditions of nonlocal nanostructures are in fact not in equilibrium, but they can be made perfect should the nonlocal bending moment be replaced by an effective nonlocal bending moment. These conclusions are substantiated, in a general sense, by other approaches in nanostructural models such as strain gradient theory, modified couple stress models and experiments.     

Thermal Buckling Analysis of Size-Dependent FG Nanobeams Based on the Third-Order Shear Deformation Beam Theory

In this paper,the thermal effects on the buckling of functionally graded(FG) nanobeams subjected to various types of thermal loading including uniform,linear and non-linear temperature changes are investigated based on the nonlocal third-order shear deformation beam theory.The material properties of FG nanobeam are supposed to vary gradually along the thickness direction according to the power-law form.The governing equations are derived through Hamilton's principle and solved analytically.Comparison examples are performed to verify the present results.Obtained results are presented for thermal buckling analysis of FG nanobeams such as the effects of the power-law index,nonlocal parameter,slenderness ratio and thermal loading in detail.     

Free torsional vibration of cracked nanobeams incorporating surface energy effects

This paper investigates surface energy effects, including the surface shear modulus, the surface stress, and the surface density, on the free torsional vibration of nanobeams with a circumferential crack and various boundary conditions. To formulate the problem, the surface elasticity theory is used. The cracked nanobeam is modeled by dividing it into two parts connected by a torsional linear spring in which its stiffness is related to the crack severity. Governing equations and corresponding boundary conditions are derived with the aid of Hamilton's principle. Then, natural frequencies are obtained analytically, and the influence of the crack severity and position, the surface energy, the boundary conditions, the mode number, and the dimensions of nanobeam on the free torsional vibration of nanobeams is studied in detail. Results of the present study reveal that the surface energy has completely different effects on the free torsional vibration of cracked nanobeams compared with its effects on the free transverse vibration of cracked nanobeams.     

Non-local free and forced vibrations of graded nanobeams resting on a non-linear elastic foundation

We consider in this paper the free and forced vibration response of simply-supported functionally graded (FG) nanobeams resting on a non-linear elastic foundation. The two-constituent Functionally Graded Material (FGM) is assumed to follow a power-law distribution through the beam thickness. Eringen׳s non-local elasticity model with material length scales is used in conjunction with the Euler–Bernoulli beam theory with von Kármán geometric non-linearity that accounts for moderate rotations. Non-linear natural frequencies of non-local FG nanobeams are obtained using He׳s Variational Iteration Method (VIM) and the direct and discretized Method of Multiple Scales (MMS), while the primary resonance analysis of an externally forced non-local FG nanobeam is performed only using the MMS. The effects of the non-local parameter, power-law index, and the parameters of the non-linear elastic foundation on the non-linear frequency-response are investigated.     

Deep postbuckling and nonlinear bending behaviors of nanobeams with nonlocal and strain gradient effects

In this paper, multi-scale modeling for nanobeams with large deflection is conducted in the framework of the nonlocal strain gradient theory and the Euler-Bernoulli beam theory with exact bending curvature. The proposed size-dependent nonlinear beam model incorporates structure-foundation interaction along with two small scale parameters which describe the stiffness-softening and stiffness-hardening size effects of nanomaterials, respectively. By applying Hamilton's principle, the motion equation and the associated boundary condition are derived. A two-step perturbation method is introduced to handle the deep postbuckling and nonlinear bending problems of nanobeams analytically. Afterwards, the influence of geometrical, material, and elastic foundation parameters on the nonlinear mechanical behaviors of nanobeams is discussed. Numerical results show that the stability and precision of the perturbation solutions can be guaranteed, and the two types of size effects become increasingly important as the slenderness ratio increases. Moreover, the in-plane conditions and the high-order nonlinear terms appearing in the bending curvature expression play an important role in the nonlinear behaviors of nanobeams as the maximum deflection increases.     

A new nonlocal bending model for Euler–Bernoulli nanobeams

This paper is concerned with the bending problem of nanobeams starting from a nonlocal thermodynamic approach. A new coupled nonlocal model, depending on two nonlocal parameters, is obtained by using a suitable definition of the free energy. Unlike previous approaches which directly substitute the expression of the nonlocal stress into the classical equilibrium equations, the proposed approach provides a methodology to recover nonlocal models starting from the free energy function. The coupled model can then be specialized to obtain a nanobeam formulation based on the Eringen nonlocal elasticity theory and on the gradient elastic model. The variational formulations are consistently provided and the differential equations with the related boundary conditions are thus derived. Nanocantilevers are solved in a closed-form and numerical results are presented to investigate the influence of the nonlocal parameters.     

Nonlocal Thermo-Electro-Mechanical coupling vibrations of axially moving piezoelectric nanobeams

This work is concerned with the thermo-electro-mechanical coupling transverse vibrations of axially moving piezoelectric nanobeams which reveal potential applications in self-powered components of biomedical nano-robot. The nonlocal theory and Euler piezoelectric beam model are employed to develop the governing partial differential equations of the mathematical model for axially moving piezoelectric nanobeams. The natural frequencies of nanobeams under simply supported and fully clamped boundary constraints are numerically determined based on the eigenvalue method. Subsequently, some detailed parametric studies are presented and it is shown that the nonlocal nanoscale effect and axial motion effect contribute to reduce the bending rigidity of axially moving piezoelectric nanobeam and hence its natural frequency decreases within the framework of nonlocal elasticity. Moreover, the natural frequency decreases with increasing the positive external voltage, axial compressive force and change of temperature, while increases with increasing the axial tensile force. The critical speed and critical axial compressive force are determined and the dynamical buckling behaviors of axially moving piezoelectric nanobeams are indicated. It is concluded the nonlocal nanoscale parameter plays a remarkable role in the size-dependent natural frequency, critical speed and critical axial compressive force.     

INVESTIGATIONS ON THE MECHANICAL CHARACTERISTICS OF NANOBEAMS BASED ON THE NONLOCAL STRAIN GRADIENT THEORY1)

基于非局部应变梯度理论,建立了一种具有尺度效应的高阶剪切变形纳米梁的力学模型. 其中,考虑了应变场和一阶应变梯度场下的非局部效应. 采用哈密顿原理推导了纳米梁的控制方程和边界条件,并给出了简支边界条件下静弯曲、自由振动和线性屈曲问题的纳维级数解. 数值结果表明,非局部效应对梁的刚度产生软化作用,应变梯度效应对纳米梁的刚度产生硬化作用,梁的刚度整体呈现软化还是硬化效应依赖于非局部参数与材料特征尺度的比值. 梁的厚度与材料特征尺度越接近,非局部应变梯度理论与经典弹性理论所预测结果之间的差异越显著.     

Electro-mechanical coupling wave propagating in a locally resonant piezoelectric/elastic phononic crystal nanobeam with surface effects

The model of a "spring-mass" resonator periodically attached to a piezoelectric/elastic phononic crystal(PC) nanobeam with surface effects is proposed, and the corresponding calculation method of the band structures is formulized and displayed by introducing the Euler beam theory and the surface piezoelectricity theory to the plane wave expansion(PWE) method. In order to reveal the unique wave propagation characteristics of such a model, the band structures of locally resonant(LR) elastic PC Euler nanobeams with and without resonators, the band structures of LR piezoelectric PC Euler nanobeams with and without resonators, as well as the band structures of LR elastic/piezoelectric PC Euler nanobeams with resonators attached on PZT-4, with resonators attached on epoxy, and without resonators are compared. The results demonstrate that adding resonators indeed plays an active role in opening and widening band gaps. Moreover, the influence rules of different parameters on the band gaps of LR elastic/piezoelectric PC Euler nanobeams with resonators attached on epoxy are discussed, which will play an active role in the further realization of active control of wave propagations.     

ANALYSIS OF NANOBRIDGE TESTS

This paper analyzes nanobridge tests with consideration of adhesive contact deformation, which occurs between a probe tip and a tested nanobeam, and deformation of a substrate or template that supports the tested nanobeam.Analytical displacement-load relation, including adhesive contact deformation and substrate deformation, is presented here for small deformation of bending.The analytic results are confirmed by finite element analysis.If adhesive contact deformation and substrate deformation are not considered in the analysis of nanobridge test data, they might lead to lower values of Young＇s modulus of tested nanobeams.     

Axisymmetric bending analysis of thick functionally graded circular plates using fourth-order shear deformation theory

In the present article, axisymmetric bending and stretching of functionally graded (FG) circular plates subjected to uniform transverse loading based on fourth-order shear deformation plate theory (FOST) have been studied. Using a fourth-order shear deformation theory, the solutions for deflection and rotation functions of FG plates are presented in terms of the corresponding quantities for a homogeneous plate using the classical plate theory (CPT), from which solutions one can easily obtain the FOST solutions for axisymmetric bending of FG circular plates. It is assumed that the effective mechanical properties of the functionally graded plates through the thickness are continuous functions of the volume fractions of the constituent parts which are themselves defined by a power-law function. Numerical results for maximum deflection and shear stress are presented for various percentages of ceramic–metal volume fractions. These results are also compared with those obtained from the first-order shear deformation plate theory of Mindlin (FST), the third-order shear deformation plate theory of Reddy (TST) as well as the exact three-dimensional elasticity solution. It is found that although the maximum deflections obtained using FOST and TST are close to each other, the through-thickness shear stress is predicted more accurately by the FOST formulation than by the TST.     

Postbuckling instability of nonlinear nanobeam with geometric imperfection embedded in elastic foundation

In this paper, the static instability of a nanobeam with geometrical imperfections that is embedded in elastic foundation is investigated. Size-dependent effect is included in the nonlinear model. It is argued that nonlocal parameter may render the nanobeam initially unstable. Static response is studied and the condition for instability is stated. The exact postbuckling solution for both the straight and curved nanobeam is presented. It is shown that the bifurcation diagram of a curved nanobeam with initial sinusoidal configuration is similar to that of a straight nanobeam in its nearest buckling mode. The results are verified with pervious relevant works on straight nanobeams and classical theory of curved beams and excellent agreement is shown.     

Towards Understanding Why the Thin Membrane Transducer Deforms:Surface Stress-Induced Buckling

This study is directed towards a comprehensive exploration on the deformation mechanism of the thin membrane transducer(TMT) caused by surface stress variation.We stress that the biomolecular interaction has changed the magnitude of the surface stress;and when the surface stress exceeds a critical value the TMT will buckle and deform.Based upon Gurtin's theory of surface elasticity and principle of finite deformation,we abstract the TMT as a nanobeam with two clamped ends,and the close-formed governing equation set is derived accordingly.A computer code via the shooting method is developed to solve the presented two-point boundary value problem.In succession,the nanobeam deflection and critical parameters for buckling are quantitatively discussed.This investigation lays the theoretical foundation of TMTs;and it is also beneficial to gain deep insight into characterizing mechanical properties of nanomaterials and engineering nano-devices.     

Effect of surface stress and surface-induced stress on behavior of piezoelectric nanobeam

A new continuum model is developed to study the influence of surface stress on the behaviors of piezoelectric nanobeams. Different from existing piezoelectric surface models which only consider the surface properties, the proposed model takes surfaceinduced initial fields into consideration. Due to the fact that the surface-induced initial fields are totally different under various boundary conditions, two kinds of beams, the doubly-clamped beam and the cantilever beam, are analyzed. Furthermore, boundary conditions can affect not only the initial state of the piezoelectric nanobeam but also the forms of the governing equations. Based on the Euler-Bernoulli beam theory, the nonlinear Green-Lagrangian strain-displacement relationship is applied. In addition, the surface area change is also considered in the proposed model. The governing equations of the doubly-clamped and cantilever beams are derived by the energy variation principle. Compared with existing Young-Laplace models, the proposed model for the doubly-clamped beam is similar to the Young-Laplace models. However, the governing equation of the cantilever beam derived by the proposed model is very different from that derived by the Young-Laplace models. The behaviors of piezoelectric nanobeams predicted by these two models also have significant discrepancies, which is owing to the surface-induced initial fields in the bulk beam.     

Bending and free vibrational analysis of bi-directional functionally graded beams with circular cross-section

The bending and free vibrational behaviors of functionally graded (FG) cylindrical beams with radially and axially varying material inhomogeneities are investigated. Based on a high-order cylindrical beam model, where the shear deformation and rotary inertia are both considered, the two coupled governing differential motion equations for the deflection and rotation are established. The analytical bending solutions for various boundary conditions are derived. In the vibrational analysis of FG cylindrical beams, the two governing equations are firstly changed to a single equation by means of an auxiliary function, and then the vibration mode is expanded into shifted Chebyshev polynomials. Numerical examples are given to investigate the effects of the material gradient indices on the deflections, the stress distributions, and the eigenfrequencies of the cylindrical beams, respectively. By comparing the obtained numerical results with those obtained by the three-dimensional (3D) elasticity theory and the Timoshenko beam theory, the effectiveness of the present approach is verified.     

LARGE DEFLECTION THEORY OF NANOBEAMS

One-dimensional nanostructured materials are often used as beams in many applications such as ultrahigh-frequency resonators and ultrasensitive sensors. Compared with usual macroscopic beams, nanobeams have much higher surface/volume ratios so that their surface energies may play a significant role. Besides, they often bear large deflections due to their typically large slenderness ratios and larger elastic ranges. There is, however, lack of a theory that takes into account of both the above two features owned by nanobeams. In this paper, we present such a theory and give applied examples to show that surface energy and large deflection may individually or jointly have notable effects.     

Influence of size effect and elastic boundary condition on the pull-in instability of nano-scale cantilever beams immersed in liquid electrolytes

In this study, the static pull-in instability of nanocantilever beams immersed in a liquid electrolyte is theoretically investigated. In modeling the nanocantilever beam, the effects of van der Waals forces, elastic boundary condition and size dependency are considered. The modified couple stress theory, containing material length scale parameter, is used to interpret the size effect which appears in micro/nanoscale structures. The modified Adomian decomposition (MAD) method is used to gain an approximate analytical expression for the critical pull-in parameters which are essential for the design of micro/nanoactuators. The results show that the beam can deflect upward or downward, based on the values of the non-dimensional parameters. It is found that the size effect greatly influences the beam deflection and is more noticeable for small thicknesses. Neglecting size effect overestimates the deflection of the nanobeam. The findings reveal that the increase of ion concentration increases the pull-in voltage but decreases the pull-in deflection. Furthermore, an increase in ion concentration increases the influence of size-dependent effect on pull-in voltage.     

用切比雪夫多项式研究短梁的弯曲计算

考虑剪切效应,利用切比雪夫多项式构造严格满足表面切应力边界条件的轴向位移表达式,建立了短梁弯曲问题的新理论．利用奇异函数把作用在短梁上的复杂外载荷表示为分布载荷,推导出了短梁弯曲时的截面正应力公式及挠曲线表达式．把采用切比雪夫多项式推导出短梁的弯曲计算公式计算结果与弹性理论计算结果进行比较,可知该方法的计算精度较高．研究结果表明：在复杂外载荷作用下,当长高比小于等于6时,剪切变形对梁的弯曲挠度影响较大,而当长高比小于3时,剪切变形对梁的弯曲应力影响较大;因此建议采用切比雪夫多项式方法给出的挠度表达式、弯曲应力进行计算,因为切比雪夫多项式方法不但给出了复杂外载荷作用下梁截面挠度、弯曲应力的计算通式,而且该方法具有计算过程简便、精度高的优点．     

梯度型非局部高阶梁理论与非局部弯曲新解法

针对文献中关于纳米结构刚度受非局部效应影响趋势的不一致预测,基于梯度型的非局部微分本构模型,利用迭代法及泰勒展开法求得了非局部高阶应力的无穷级数表达,非局部应力相当于经典弯曲应力与非局部挠度的逐阶梯度之和. 然后推导了梯度型非局部高阶梁弯曲的挠曲轴微分方程,并结合正则摄动思想,求解了非局部挠度的表达式. 最后给出数值算例,具体量化挠度受非局部尺度因子的影响大小. 研究表明:相比于其经典值,纳米结构的非局部弯曲挠度可呈现出或增大或减小或不变的趋势,考虑梯度型非局部高阶应力降低或提高或不影响纳米结构的刚度,具体结果依赖于外载和边界约束的类型. 算例显示外载形式和边界约束条件均各自独立地影响着纳米结构的非局部弯曲挠度,同时首次观察到非局部最大弯曲挠度的位置可能受非局部尺度因子的影响. 研究结论解决了非局部弹性力学应用于纳米结构的若干疑点,并为理论的发展和优化提供支持.      

Nonlinear resonance responses of geometrically imperfect shear deformable nanobeams including surface stress effects

In this work, a thorough investigation is presented into the nonlinear resonant dynamics of geometrically imperfect shear deformable nanobeams subjected to harmonic external excitation force in the transverse direction. To this end, the Gurtin–Murdoch surface elasticity theory together with Reddy’s third-order shear deformation beam theory is utilized to take into account the size-dependent behavior of nanobeams and the effects of transverse shear deformation and rotary inertia, respectively. The kinematic nonlinearity is considered using the von Kármán kinematic hypothesis. The geometric imperfection as a slight curvature is assumed as the mode shape associated with the first vibration mode. The weak form of geometrically nonlinear governing equations of motion is derived using the variational differential quadrature (VDQ) technique and Lagrange equations. Then, a multistep numerical scheme is employed to solve the obtained governing equations in order to study the nonlinear frequency–response and force–response curves of nanobeams. Comprehensive studies into the effects of initial imperfection and boundary condition as well as geometric parameters on the nonlinear dynamic characteristics of imperfect shear deformable nanobeams are carried out through numerical results. Finally, the importance of incorporating the surface stress effects via the Gurtin–Murdoch elasticity theory, is emphasized by comparing the nonlinear dynamic responses of the nanobeams with different thicknesses.