Geometrically nonlinear analysis of Timoshenko piezoelectric nanobeams with flexoelectricity effect based on Eringen's differential model

This study analyzes the nonlinear free vibration and post-buckling of nanobeams with flexoelectric effect based on Eringen's differential model. The nanobeam is modeled based on Timoshenko beam's theory. The von-Kármán strain–displacement relation together with the electrical Gibbs free energy and Hamilton's principle are employed to derive equations of motion. The nonlinear free vibration frequencies are obtained for pinned–pinned (P–P) and clamped–clamped (C–C) boundary conditions. Multiple scales method is employed to obtain the closed-form solution for the nonlinear governing equations. By employing this methodology, the natural frequencies of nanobeams are obtained and their post-buckling behavior is examined. The influence of nonlocal parameter, amplitude ratio, and input voltage on the top surface and flexoelectricity constant on nonlinear free vibration and post-buckling characteristics of nanobeam is investigated. In this paper, it is concluded that the flexoelectricity has a significant effect on free vibration of the beams in nano-scale and its effect has to be considered in designing nano-electro-mechanical systems (NEMS) such as nano- generators and nano-sensors.     

Peridynamics model for flexoelectricity and damage

A flexoelectric peridynamic (PD) theory is proposed. In the PD framework, the formulation introduces a nanoscale flexoelectric coupling that entails non-uniform strain in centrosymmetric dielectrics. This potentially enables PD modeling of a large class of phenomena in solid dielectrics involving cracks, discontinuities etc. wherein large strain gradients are present and the classical electromechanical theory based on partial differential equations do not directly apply. PD electromechanical equations, derived from Hamilton's principle, satisfy the global balance laws. Linear PD constitutive equations reflect the electromechanical coupling effect, with the mechanical force state affected by the polarization state and the electrical force state in turn by the displacement state. An analytical solution to the PD electromechanical equations is presented for the static case when a point mechanical force and a point electric force act in an infinite 3D solid dielectric. A parametric study on how different length scales influence the response is undertaken. In addition, the model is extended to incorporate damage using phase field – an order parameter, supplemented with a PD bond breaking criterion to study flexoelectric effects in damage and fracture problems. To demonstrate the performance of our proposal, we first simulate, considering small flexoelectricity effect and no damage, an externally pressured 2D flexoelectric disk subjected to a potential difference between the inner and outer surfaces and compare the results with existing solutions in the literature. Next, we simulate a plate with a central pre-crack under tension considering damage and flexoelectricity effects, and study the effect of various constitutive parameters on the damage evolution. We also furnish a classical derivation of phase field based flexoelectricity in Appendix I.     

基于非局部弹性应力场理论的纳米尺度效应研究:纳米梁的平衡条件、控制方程以及静态挠度

该文成功地解答了3个关于非局部应力理论用于纳米梁的问题:(ⅰ) 在绝大多数研究中,非局部效应增加导致纳米结构体刚度下降,其现象表现为弯曲挠度增加,固有频率减少,屈曲载荷下降,但为什么Eringen 的非局部弹性理论给出了完全相反的结论;(ⅱ) 为什么在某些研究结果中,非局部效应消失或是对研究结果无影响,比如纳米悬臂梁在集中载荷作用下的弯曲挠度; (ⅲ) 在高阶控制方程中,为什么高阶边界条件不存在．通过应用非局部弹性理论和精确变分原理分析纳米梁的弯曲问题,推导出全新的平衡条件、控制方程、边界条件和静态响应．这些方程和条件包含了与之前的相关研究结果符号相反的高阶微分项,这一差别导致了纳米效应对结构体的影响结果完全相反． 还证明之前为大家所公认的纳米梁静态或动态平衡条件实际上没有达到平衡,只有用等效弯矩代替非局部弯矩时,才可达到平衡．这些结论通常是可以被其它方法,比如应变梯度理论、耦合应力模型以及相关实验所证明．     

Nonlinear secondary resonance of nanobeams under subharmonic and superharmonic excitations including surface free energy effects

The main objective of this study is to predict both the subharmonic and superharmonic resonances of the nonlinear oscillation of nanobeams in the presence of surface free energy effects. To this purpose, Gurtin–Murdoch elasticity theory is adopted to the classical beam theory in order to consider the surface Lame constants, surface mass density, and residual surface stress within the differential equations of motion. The Galerkin method together with the method of multiple scales is utilized to investigate the size-dependent response of nanobeams under hard excitations corresponding to various boundary conditions. A parametric analysis is carried out to indicate the influence of the surface elastic parameters on the frequency-response as well as amplitude-response of the nonlinear secondary resonance including multiple vibration modes and interactions between them. It is seen that for the superharmonic excitation, except for the clamped–free boundary condition, the jump phenomenon is along the hardening direction, while in the clamped–free end supports, it is along the softening direction. Moreover, it is revealed that for the subharmonic excitation, within a specific range of the excitation amplitude, the nanobeam is excited, and this range shifts to lower external force by incorporating the surface free energy effects. It is found that in the case of superharmonic excitation, the value of the excitation frequency associated with the bifurcation point at the peak of the frequency-response curve increases by taking the surface free energy effect into consideration.     

A nonlocal higher-order model including thickness stretching effect for bending and buckling of curved nanobeams

An analytical approach for static bending and buckling analyses of curved nanobeams using the differential constitutive law, consequent to Eringen’s strain-driven integral model coupled with a higher-order shear deformation accounting for through thickness stretching is presented. The formulation is general in the sense that it can be deduced to examine the influence of different structural theories, for static and dynamic analyses of curved nanobeams. The governing equations derived using Hamiltons principle are solved in conjunction with Naviers solutions. The formulation is validated considering problems for which solutions are available. A comparative study is made here by various theories obtained through the formulation. The effects various structural parameters such as thickness ratio, beam length, rise of the curved beam, and nonlocal scale parameter are brought out on bending and stability characteristics of curved nanobeams.     

Exact modes for post-buckling characteristics of nonlocal nanobeams in a longitudinal magnetic field

An exact mode solution that investigates the prebuckling and postbuckling characteristics of nonlocal nanobeams with fixed–fixed, hinged–hinged, and fixed–hinged boundary conditions in a longitudinal magnetic field is determined. The geometric nonlinearity arising from mid-plane stretching is considered to obtain the nonlinear governing equation of motion by virtue of Hamilton's principle. The influences of the nonlocal and magnetic parameters on the prebuckling and postbuckling dynamics of nanobeams with various boundary conditions are evaluated, indicating that the critical buckling force can be decreased with the increase of the nonlocal parameter while can be increased with increasing the magnetic parameter. It is demonstrated that the first natural frequency of the nanobeam with fixed–fixed and fixed–hinged conditions in postbuckling configuration is increased from zero to a constant value for higher values of the nonlocal parameter with increasing the axial force. The second natural frequency of the buckled nanobeam is always decreased with an increase of the nonlocal parameter. The results show that the internal resonance between the first and second modes of the postbuckling nanobeams can be quickly and easily activated by increasing the nonlocal parameters, especially for fixed–fixed and hinged–hinged boundary conditions. In addition, the results obtained by exact mode solution are compared those obtained by classical mode solution. It is found that the classical mode is valid only for nonlocal nanobeams with the hinged–hinged boundary conditions.     

A flexoelectric spherical microshell model incorporating the strain gradient effect

In this paper, a size-dependent flexoelectric spherical microshell model is proposed considering flexoelectric effect and strain gradient effect. By means of the variation principle, explicit expressions of the governing equations and the boundary conditions are deduced. Solving corresponding governing equations, analytical solutions of both direct and converse flexoelectric responses in static axisymmetric bending problem are obtained. Then, the flexoelectric responses in barium strontium titanate spherical microshells with and without a circular top opening are numerically investigated. Both the direct and converse flexoelectric responses are found to vary non-monotonically as the central angle increases. The converse flexoelectric bending is examined to exist even in clamped spherical microshells, which is different from the case of flat structures. In addition, for all cases, the strain gradient effect will highly reduce the flexoelectric responses, particularly when the thickness approaches the material internal scale constants.     

A mixed differential quadrature and finite element free vibration and buckling analysis of thick beams on two-parameter elastic foundations

As a first endeavor, a mixed differential quadrature (DQ) and finite element (FE) method for boundary value structural problems in the context of free vibration and buckling analysis of thick beams supported on two-parameter elastic foundations is presented. The formulations are based on the two-dimensional theory of elasticity. The problem domain along axial direction is discretized using finite elements. The resulting system of equations and the related boundary conditions are discretized in the thickness direction and in strong-form using DQM. The method benefits from low computational efforts of the DQ in conjunction with the effectiveness of the FE method in general geometry and systematic boundary treatment resulting in highly accurate and fast convergence behavior solution. The boundary conditions at the top and bottom surface of the beams are implemented accurately. The presented formulations provide an effective analysis tool for beams free of shear locking. Comparisons are made with results from elasticity solutions as well as higher-order beam theory.     

Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects

In this paper, the linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams are studied based on the Gurtin–Murdoch surface stress theory. Firstly, the constitutive equations of fractional viscoelasticity theory are considered, and based on the Gurtin–Murdoch model, stress components on the surface of the nanobeam are incorporated into the axial stress tensor. Afterward, using Hamilton's principle, equations governing the two-dimensional vibrations of fractional viscoelastic nanobeams are derived. Finally, two solution procedures are utilized to describe the time responses of nanobeams. In the first method, which is fully numerical, the generalized differential quadrature and finite difference methods are used to discretize the linear part of the governing equations in spatial and time domains. In the second method, which is semi-analytical, the Galerkin approach is first used to discretize nonlinear partial differential governing equations in the spatial domain, and the obtained set of fractional-order ordinary differential equations are then solved by the predictor–corrector method. The accuracy of the results for the linear and nonlinear vibrations of fractional viscoelastic nanobeams with different boundary conditions is shown. Also, by comparing obtained results for different values of some parameters such as viscoelasticity coefficient, order of fractional derivative and parameters of surface stress model, their effects on the frequency and damping of vibrations of the fractional viscoelastic nanobeams are investigated.     

A general nonlocal nonlinear model for buckling of nanobeams

This study presents a unified model for the nonlocal response of nanobeams in buckling and postbuckling states. The formulation is suitable for the classical Euler–Bernoulli, first-order Timoshenko, and higher-order shear deformation beam theories. The small-scale effect is modeled according to the nonlocal elasticity theory of Eringen. The equations of equilibrium are obtained using the principle of virtual work. The stress resultants are developed taking into account the nonlocal effect. Analytical solutions for the critical buckling load and the amplitude of the static nonlinear response in the postbuckling state are obtained. It is found out that as the nonlocal parameter increases, the critical buckling load reduces and the amplitude of buckling increases. Numerical results showing variation of the critical buckling load and the amplitude of buckling with the nonlocal parameter and the length-to-height ratio for simply supported and clamped–clamped nanobeams are presented.     

Surface effects on nonlinear free vibration of functionally graded nanobeams using nonlocal elasticity

In this paper, to consider all surface effects including surface elasticity, surface stress, and surface density, on the nonlinear free vibration analysis of simply-supported functionally graded Euler–Bernoulli nanobeams using nonlocal elasticity theory, the balance conditions between FG nanobeam bulk and its surfaces are considered to be satisfied assuming a cubic variation for the component of the normal stress through the FG nanobeam thickness. The nonlinear governing equation includes the von Kármán geometric nonlinearity and the material properties change continuously through the thickness of the FG nanobeam according to a power-law distribution of the volume fraction of the constituents. The multiple scale method is employed as an analytical solution for the nonlinear governing equation to obtain the nonlinear natural frequencies of FG nanobeams. The effect of the gradient index, the nanobeam length, thickness to length ratio, mode number, amplitude of deflection to radius of gyration ratio and nonlocal parameter on the frequency ratios of FG nanobeams is investigated.     

Frequency analysis of carbon and silicon nanosheet with surface effects

Silicon-based microelectromechanical system (MEMS) and nanoelectromechanical systems (NEMS) have been used to design and fabricate sensitive sensors and actuators. Recent research trends show that graphene and carbon nanotubes (CNTs) have been used to change the surface properties of silicon-based MEMS and NEMS to improve different mechanical, optical and electrical properties of silicon-based composites. In this paper, we focus on analyzing the vibrational characteristics of silicon-based devices when the surface of silicon is coated with single-layer graphene and horizontally aligned carbon nanotubes (HACNTs). To perform the analysis, we use multi-scale finite element approach for developing graphene–silicon nanocomposites (GSNCs) and carbon nanotube-silicon nanocomposites (CSNC) composites in which interface layer of silicon with graphene or CNT is modeled using bonded contact element. Subsequently, we performed modal analysis to find the first transverse mode frequency of GSNC and CSNC composites for beam with smaller as well as longer lengths. The numerical model is compared with classical beam theory with and without surface effect. For GSNCs composites, we take a fixed-free case with lengths in the range of (20 Å–120 Å) and (400 Å–2000 Å), respectively. For CSNC composites, CNT diameter is varied from (5 Å–30 Å) for single walled nanotube. Subsequently, we analyze the influence of HACNTs-on-silicon on its vibrational characteristics. The analysis presented in the paper demonstrate that GSNCs offer a higher bending stiffness compared to single layer graphene (SLGs) and isolated silicon nanosheet which lead to higher natural frequency. A similar trend is found in the case of HACNTs on silicon NS when the number of tubes increases.     

Three-dimensional layerwise-finite element free vibration analysis of thick laminated annular plates on elastic foundation

Using a three-dimensional layerwise-finite element method, the free vibration of thick laminated circular and annular plates supported on the elastic foundation is studied. The Pasternak-type formulation is employed to model the interaction between the plate and the elastic foundation. The discretized governing equations are derived using the Hamilton’s principle in conjunction with the layerwise theory in the thickness direction, the finite element (FE) in the radial direction and trigonometric function in the circumferential direction, respectively. The fast rate of convergence of the method is demonstrated and to verify its accuracy, comparison studies with the available solutions in the literature are performed. The effects of the geometrical parameters, the material properties and the elastic foundation parameters on the natural frequency parameters of the laminated thick circular and annular plates subjected to various boundary conditions are presented.     

Size-dependent pull-in phenomena in electrically actuated nanobeams incorporating surface energies

A modified continuum model of electrically actuated nanobeams is presented by incorporating surface elasticity in this paper. The classical beam theory is adopted to model the bulk, while the bulk stresses along the surfaces of the bulk substrate are required to satisfy the surface balance equations of the continuum surface elasticity. On the basis of this modified beam theory the governing equation of an electrically actuated nanobeam is derived and a powerful technology, analog equation method (AEM) is applied to solve this complex problem. Beams made from two materials: aluminum and silicon are chosen as examples. The numerical results show that the pull-in phenomena in electrically actuated nanobeams are size-dependent. The effects of the surface energies on the static and dynamic responses, pull-in voltage and pull-in time are discussed.     

ADFE METHOD WITH HIGH ACCURACY FOR NONLINEAR PARABOLIC INTEGRO-DIFFERENTIAL SYSTEM WITH NONLINEAR BOUNDARY CONDITIONS

Alternating direction finite element (ADFE) scheme for d-dimensional nonlin-ear system of parabolic integro-differential equations is studied. By using a local approxi-mation based on patches of finite elements to treat the capacity term qi(u), decomposition of the coefficient matrix is realized; by using alternating direction, the multi-dimensional problem is reduced to a family of single space variable problems, calculation work is sim-plified; by using finite element method, high accuracy for space variant is kept; by using inductive hypothesis reasoning, the difficulty coming from the nonlinearity of the coeffi-cients and boundary conditions is treated; by introducing Ritz-Volterra projection, the difficulty coming from the memory term is solved. Finally, by using various techniques for priori estimate for differential equations, the unique resolvability and convergence proper-ties for both FE and ADFE schemes are rigorously demonstrated, and optimal H1 and L2 norm space estimates and O((△t)2) estim     

Vibration of nanobeams of different boundary conditions with multiple cracks based on nonlocal elasticity theory

The aim of this paper is to study the free vibration of nanobeams with multiple cracks. The analysis procedure is based on nonlocal elasticity theory. This theory states that stress at a point is a function of strains at all points in the continuum. The nonlocal elasticity theory becomes significant for small length scale in micro and nanostructures. The effects of nonlocality, crack location and crack parameter are investigated on the natural frequencies of the cracked nanobeam. In this study, analytical solutions are given for cracked Euler–Bernoulli nanobeams of different boundary conditions.     

非局部因子和表面效应对微纳米材料振动特性的影响

基于非局部理论和表面效应模型,导出表面吸附物对微纳米材料的动力学方程,研究非局部因子和表面能对微纳米传感器振动特性的影响．结果显示,非局部因子、表面能、吸附物种类、附加刚度和基底种类对微纳米结构的振动特性有重要影响．     

Numerical Methods and Volatility Models for Valuing Cliquet Options

Several numerical issues for valuing cliquet options using PDE methods are investigated. The use of a running sum of returns formulation is compared to an average return formulation. Methods for grid construction, interpolation of jump conditions, and application of boundary conditions are compared. The effect of various volatility modelling assumptions on the value of cliquet options is also studied. Numerical results are reported for jump diffusion models, calibrated volatility surface models, and uncertain volatility models.     

Static response and free vibration analysis of FGM plates using higher order shear deformation theory

Free vibration and static analysis of functionally graded material (FGM) plates are studied using higher order shear deformation theory with a special modification in the transverse displacement in conjunction with finite element models. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. The fundamental equations for FGM plates are derived using variational approach by considering traction free boundary conditions on the top and bottom faces of the plate. Results have been obtained by employing a continuous isoparametric Lagrangian finite element with 13 degrees of freedom per node. Convergence tests and comparison studies have been carried out to demonstrate the efficiency of the present model. Numerical results for different thickness ratios, aspect ratios and volume fraction index with different boundary conditions have been presented. It is observed that the natural frequency parameter increases for plate aspect ratio, lower volume fraction index n and smaller thickness ratios. It is also observed that the effect of thickness ratio on the frequency of a plate is independent of the volume fraction index. For a given thickness ratio non-dimensional deflection increases as the volume fraction index increases. It is concluded that the gradient in the material properties plays a vital role in determining the response of the FGM plates.     

Vibration monitoring of cylindrical shells using piezoelectric sensors

From linear vibration theory for beams and plates, one can express the response as a linear combination of its natural modes. For beams, these eigenfunctions can be shown to be mutually orthogonal for any boundary conditions. For plates, orthogonality of the modes is not guaranteed, but can be proven for various boundary conditions. Modal analysis for beams and plates allows the system response to be broken down into simpler vibration models, due to the orthogonality of the modes. Here the modal analysis approach is extended to the vibration of thin cylindrical shells. The longitudinal, radial, and circumferential displacements are coupled with each other, due to Poisson's ratio and the curvature of the shell. As will be shown, the mode shapes can be solved analytically with numerically determined coefficients. The immediate application of this work will be for modal sensing of cylindrical shell vibrations using thin piezoelectric films.