Bending Coupling Research Articles

Material characterisation of nanowires with intrinsic stress

When fabricating nanowires (NWs) in a doubly-clamped beam configuration it is possible for a residual axial stress to be generated. Here, we show that material characterisation of metal and semiconductor NWs subjected to residual axial stress can be problematic. Benchmark measurements of the Young’s modulus of NWs are performed by sectioning a doubly-clamped NW into two cantilevered wires, eliminating residual axial stress. Use of models for doubly-clamped beams that incorporate the effects of residual stress are found to lead to ambiguity in the extracted Young’s modulus as a function of displacement fit range, even for NWs with no residual stress. This is due to coupling of bending and axial stress effects at small displacements, and the limited displacement range of force curves prior to fracture or plastic deformation. This study highlights the importance of fabricating metal and semiconductor NWs that exhibit little or no residual axial stress for materials characterisation.

Nonlinear vibration analysis of functionally graded beams considering the influences of the rotary inertia of the cross section and neutral surface position

ABSTRACTIn the paper work, the nonlinear vibration response of functionally graded (FG) Euler–Bernoulli beam resting on elastic foundation is studied. Based on von Kármán’s geometric nonlinearity, the partial differential governing equations describing the nonlinear vibration of FG Euler–Bernoulli beam are derived from Hamilton’s principle and are reduced to an ordinary nonlinear differential equation with quadratic and cubic nonlinear terms via Galerkin’s procedure. Due to unsymmetrical material variation along the thickness of FG beam, the neutral surface concept is proposed to remove the stretching and bending coupling effect and the rotary inertia of the cross section is incorporated to obtain an analytical solution. Numerical results are presented to show the effects of the nonlocal parameters and vibration amplitude on the frequency responses. This results may be useful in design and engineering applications.

Boundary Layer Modeling of Nonlinear Axial Buckling Behavior of Functionally Graded Cylindrical Nanoshells Based on the Surface Elasticity Theory

The main purpose of the present study is to examine the axial buckling and postbuckling response of cylindrical nanoshells made of functionally graded (FG) materials in the presence of surface free energy effects. To accomplish this purpose, an efficient size-dependent shell model is introduced through combination of the classical shell theory with the well-known Gurtin–Murdoch surface elasticity theory. The volume fraction of FG nanoshells made of the mixture of aluminum and silicon is defined using a power law function. Also, in order to eliminate the stretching–bending coupling terms, the change in the position of physical neutral plane corresponding to different volume fractions is taken into account. Based upon the principle of virtual work, the non-classical governing differential equations and boundary conditions are derived. Subsequently, the governing equations are deduced to the boundary layer-type equations incorporating simultaneously the nonlinear large deflections and size effect. Finally, a perturbation-based solving process is put to use to predict the size-dependent critical buckling loads and related postbuckling equilibrium curves for FG nanoshells with various values of material property gradient index and shell thickness. It is found that the surface free energy effect is more prominent for FG nanoshell with lower shell thickness and higher material property gradient index.

Coupled bending and torsional vibration of a rotor system with nonlinear friction

Unacceptable vibrations induced by the nonlinear friction in a rotor system seriously affect the health and reliability of the rotating machinery. To find out the basic excitation mechanism and characteristics of the vibrations, a coupled bending and torsional nonlinear dynamic model of rotor system with nonlinear friction is presented. The dynamic friction characteristic is described with a Stribeck curve, which generates nonlinear friction related to relative velocity. The motion equations of unbalance rotor system are established by the Lagrangian approach. Through numerical calculation, the coupled vibration characteristics of a rotor system under nonlinear friction are well investigated. The influence of main system parameters on the behaviors of the system is discussed. The bifurcation diagrams, waterfall plots, the times series, orbit trails, phase plane portraits and Poincare maps are obtained to analyze dynamic characteristics of the rotor system and the results reveal multiform complex nonlinear dynamic responses of rotor system under rubbing. These analysis results of the present paper can effectively provide a theoretical reference for structural design of rotor systems and be used to diagnose selfexcited vibration faults in this kind of rotor systems. The present research could contribute to further understanding on the self-excited vibration and the bending and torsional coupling vibration of the rotor systems with Stribeck friction model.

Bending and stability analysis of size-dependent compositionally graded Timoshenko nanobeams with porosities

In this article, static deflection and buckling of functionally graded (FG) nanoscale beams made of porous material are carried out based on the nonlocal Timoshenko beam model which captures the small scale influences. The exact position of neutral axis is fixed, to eliminate the stretching and bending coupling due to the unsymmetrical material change along the FG nanobeams thickness. The material properties of FG beam are graded through the thickness on the basis of the power-law form, which is modified to approximate the material properties with two models of porosity phases. By employing Hamilton's principle, the nonlocal governing equations of FG nanobeams are obtained and solved analytically for simply-supported boundary conditions via the Navier-type procedure. Numerical results for deflection and buckling of FG nanoscale beams are presented and validated with those existing in the literature. The influences of small scale parameter, power law index, porosity distribution and slenderness ratio on the static and stability responses of the FG nanobeams are all explored.

An equivalent model of corrugated panels with axial and bending coupling

In this paper, modification of an existing equivalent model of the corrugated panel is investigated. The axial and bending coupling of the corrugated panel supplements the previous equivalent properties when the corrugated panel has a fixed boundary condition. The analytical expressions of the coupling vertical deflections are obtained and verified by the finite element method. A method to eliminate the vertical deflection is proposed, and the importance of the coupling effect is demonstrated by the application of the modified model in a compliant structure.

A review on energy harvesting from ocean waves by piezoelectric technology

AbstractPiezoelectric materials have been widely used to harvest energy from ocean waves. This review is to introduce and review the development of the technologies. First, from comparison of the three major energy conversion techniques namely electrostatic, electromagnetic and piezoelectric technologies, in terms of power generation capability, transmission efficiency, and structural installation and economic costs, the advantages of applying piezoelectric energy conversion technology are identified. Second, the review sums up different methodologies and designs of harvesting energy from ocean waves based on different piezoelectric effects. In particular, the designs and efficiencies of available harvesters based on the piezoelectric effects from longitudinal, bending, and shear couplings are introduced and discussed. Finally, the futuristic research directions and methods for improving the efficiencies of the piezoelectric harvesters are discussed profoundly in this area.

基于梁格法的小半径钢筋混凝土弯箱梁桥空间受力特性分析

弯箱梁桥造型优美，能很好地适应和改善道路线型，近年来被广泛地应用于城市桥梁中。但弯箱梁受力复杂，需采用空间分析方法进行分析。本文采用MIDAS/Civil建立全桥梁格模型，分析了小半径弯箱梁结构的空间受力特性，研究了其在荷载作用下的位移以及支座反力特性。通过分析空间模型的计算结果，小半径弯箱梁结构存在明显的弯扭耦合效应，并可引起弯箱梁截面的扭转以及支座反力的不均匀等现象，通过设置支座预偏心可有效消减支座反力不均匀现象。 Curved box girder bridge is beautiful, can adapt to and improve the road linearity. In recent years, it has been widely used in urban bridges. But the force of curved box girder bridge is complex, it needs to use the spatial analysis method for analysis. In this paper, the full-bridge grid model was established by MIDAS/Civil based on grillage method, and the spatial force characteristics of the small radius curved box girder were analyzed, and the displacement and bearing reaction characteristics under the load were studied. By analyzing the calculation results of the space model, it can be seen that there are obvious bending and torsional coupling effects of the small radius curved box girder structure, which can cause the torsion of the cross section of the bending box and the nonuniformity of the bearing reaction. However, there is an effective reduction of the nonuniformity of the bearing reaction by bearing eccentricity.

Analytical prediction of coupling losses in bend silicon subwavelength optical interconnect

For the first time, singular perturbation technique in conjunction with effective index method is used to quantitatively estimate parametric variations in a bend rectangular waveguide of subwavelength dimensions that forms a basic unit of photonics integrated circuits optical interconnect. These parametric variations are correlated to losses due to bending and coupling of bend and straight waveguides. The results are compared with that obtained by more rigorous finite elements method. A scheme to compensate the losses is also proposed. In case of tight bends (bending radius ≈ 10 μm) coupling loss compensation of about 40 dB is achieved.

General analytical sensitivity analysis of composite laminated plates and shells for classical and first-order shear deformation theories

The paper describes a general analytical sensitivity analysis method for the composite laminated plates and shells, which is applied to both classical and first-order shear deformation theories and based on the finite element methods. The first derivatives of engineering constants of lamina with respect to fibre volume fractions are computed based on the micromechanics theory. The first derivatives of extensional, bending and bending-extensional coupling stiffnesses of laminate with respect to fibre volume fractions and fibre orientations are derived using classical and first order theories of laminate. Computational methods for the first derivatives of total stiffness and mass matrices are proposed based on finite element method. The analytical sensitivity analysis methods for static responses, eigenvalues and eigenvectors are described using the total stiffness and mass matrices, and their first derivatives. Sensitivity analyses of composite laminated plate and shell are demonstrated. The merits and disadvantages are also discussed.

A deformation mechanism based material model for topology optimization of laminated composite plates and shells

This paper presents a novel deformation mechanism based material model for topology optimization of laminated plates and shells considering large displacements. Discussed firstly are the one-node hinges in optimum designs of plate and shell structures and the numerical issues caused by void elements in geometrical nonlinear analysis. To circumvent these two problems, we propose a new material model in which different penalties are applied to different strain energy terms related to extensional, shear, bending and extensional-bending coupling deformation mechanisms and void elements are removed in nonlinear finite element analysis. An efficient algorithm is developed by using the present material model and the moving iso-surface threshold method. Numerical results are presented for isotropic and composite plates and shells and compared with those available in the literature to validate the present material model.

Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation

This article presents closed-form solutions for the frequency analysis of rectangular functionally graded material (FGM) thin plates subjected to initially in-plane loads and with an elastic foundation. Based on classical thin plate theory, the governing differential equations are derived using Hamilton’s principle. A neutral surface is used to eliminate stretching–bending coupling in FGM plates on the basis of the assumption of constant Poisson’s ratio. The resulting governing equation of FGM thin plates has the same form as homogeneous thin plates. The separation-of-variables method is adopted to obtain solutions for the free vibration problems of rectangular FGM thin plates with separable boundary conditions, including, for example, clamped plates. The obtained normal modes and frequencies are in elegant closed forms, and present formulations and solutions are validated by comparing present results with those in the literature and finite element method results obtained by the authors. A parameter study reveals the effects of the power law index n and aspect ratio a/b on frequencies.

Axial–bending coupling vibration of mass eccentric double-beam system with discrete elastic connections

The existence of mass eccentricity will lead to the energy transfer between axial and flexural vibrations of a beam. To study the coupling properties of a double-Timoshenko beam system, a non-uniform coupled double-beam system is modeled in which the upper beam is typical and the lower beam is mass eccentric simulated by a non-uniform two-layer Timoshenko beam. By incorporating Hamilton’s principle and spectral element method, the axial–bending coupled governing equations of the system are derived and the approach can also be easily used to analyze the influences of the parameters and other coupled beam systems. Both the free and forced vibration results of a double-beam system by this method are consistent with the corresponding finite element model’s and thus this method is validated. The coupled properties and their mechanism are revealed. The influences of axial and transverse flexible connection on the coupling properties including free and forced vibration are investigated. A systematic matching principle of reducing the vibration of the coupled system is proposed.

Zero-frequency and extremely slow elastic edge waves in mechanical granular graphene

We report here the theoretical description and analysis of edge elastic waves in a semi-infinite mechanical granular graphene structure. The granular graphene, composed of spherical beads arranged in a single-layer honeycomb structure, is studied for two types of edge configurations: zigzag and armchair. Due to the existence of the rotational degrees of freedom of the grains, rotation-associated shear, bending and torsional couplings between the neighbor beads are activated. The dispersion curves of the edge waves are theoretically derived and numerically analyzed for various configurations of bead couplings, as well as the existence of edge states when the torsional or/and bending rigidities are weak/vanishing. Quasi-flat edge mode dispersion curves with near zero frequency are observed for both the zigzag and armchair edges. These quasi-flat dispersion curves, supporting the propagation of waves with extremely slow group velocity, tend to be perfect zero-frequency modes for zero torsional rigidity or vanish for zero bending rigidity, indicating that weak bending and torsional interbead interactions are critical in the transformation of zero-frequency modes into extremely slow propagating modes. These results on edge waves in mechanical granular graphene structures with rotational degrees of freedom are the necessary preliminary step for the design of granular meta-graphenes with artificial symmetry breaking for inducing topologically protected unidirectional edge states.

Analysis of flexoelectric response in nanobeams using nonlocal theory of elasticity

This paper is concerned with the derivation of exact solutions for the static responses of simply supported nonlocal flexoelectric nanobeams. Considering both the direct and the converse flexoelectric effects, and employing the nonlocal theory of elasticity, the governing equations and the associated boundary conditions of the beams are derived to obtain the exact solutions for the displacements, nonlocal stresses and the electric potential in the beams. Both the direct and the converse flexoelectric effects are influenced by the nonlocal parameter. Active beams significantly counteract the applied mechanical load by virtue of the converse flexoelectric effect. The normal and the transverse shear deformations in the beams are affected by the converse flexoelectric effect resulting in the coupling of bending and stretching deformations in the beams even if the beams are homogeneous. Because of the consideration of the nonlocal theory of elasticity, the nonlocal stiffness of the beam appears to be less than the classical stiffness of the beam. The nonlocal elasticity does not influence the stresses of the passive beam while the nonlocal stresses in the active beam due to converse flexoeletric effect are less than the local or classical stresses in the active beam. The benchmark results presented here may be useful for verifying the numerical model and experimental results for nonlocal flexoelectric nanobeams. The present study suggests that the flexoelectric nanobeams may be effectively exploited for developing advanced smart nanosensors and nanoactuators. The research work carried out here also conveys that the nonlocal theory of elasticity must be employed for accurate analysis of flexoelectric solids.

Bifurcation analysis for vibrations of a turbine blade excited by air flows

A reduced three-degree-of-freedom model simulating the fluid-structure interactions (FSI) of the turbine blades and the oncoming air flows is proposed. The equations of motions consist of the coupling of bending and torsion of a blade as well as a van der Pol oscillation which represents the time-varying of the fluid. The 1:1 internal resonance of the system is analyzed with the multiple scale method, and the modulation equations are derived. The two-parameter bifurcation diagrams are computed. The effects of the system parameters, including the detuning parameter and the reduced frequency, on responses of the structure and fluid are investigated. Bifurcation curves are computed and the stability is determined by examining the eigenvalues of the Jacobian matrix. The results indicate that rich dynamic phenomena of the steady-state solutions including the saddle- node and Hopf bifurcations can occur under certain parameter conditions. The parameter region where the unstable solutions occur should be avoided to keep the safe operation of the blades. The analytical solutions are verified by the direct numerical simulations.

Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach

In this study, a novel size-dependent beam model is proposed for nonlinear free vibration of a functionally graded (FG) nanobeam with immovable ends based on the nonlocal strain gradient theory (NLSGT) and Euler-Bernoulli beam theory in conjunction with the von-Kármán's geometric nonlinearity. It is assumed the material properties of the nanobeam changes continuously in the thickness direction according to simple power-law form. To remove the stretching and bending coupling due to the unsymmetrical material variation along the thickness, the formulation of the problem is developed based on a new reference surface. The Hamilton's principle is utilized to derive the equations of the motion and the corresponding boundary conditions. The partial nonlinear differential equation describes the nonlinear vibration of FG nanobeam is reduced to an ordinary nonlinear differential equation with cubic nonlinearity via Galerkin's approach under the assumption that the axial inertia is negligible. A closed-form solution is obtained for nonlinear frequency by the novel Hamiltonian approach, and some illustrative numerical examples are given in order to study the effects of the strain gradient length scale, the nonlocal parameters, vibration amplitude and various material compositions on the ratio of nonlinear frequency to linear frequency (the nonlinear frequency ratio).

Analysis of smart nanobeams integrated with a flexoelectric nano actuator layer

This paper is concerned with the derivation of exact solutions for the static responses of simply supported nanobeams integrated with a flexoelectric layer acting as the distributed nano actuator. Considering both the direct and the converse flexoelectric effects, the governing equations and the associated boundary conditions of the overall beams are derived to obtain exact solutions for the displacements and the electric potential in the flexoelectric layer and the substrate beam. Due to the converse flexoelectric effect, the active flexoelectric layer significantly counteracts the deformations of the substrate beams caused by the applied mechanical load, resulting in the coupling of bending and stretching deformations in the substrate beams. For particular values of the length and thickness of the substrate beam and the applied voltage in the flexoelectric layer, the deflection of the substrate beam due to the converse flexoelectric effect increases with increasing thickness of the flexoelectric layer up to a certain value of the latter. Beyond this value, any further increase in the thickness of the flexoelectric layer causes a decrease in its ability to actuate the substrate beam. The electric potential varies linearly across the thickness of the flexoelectric layer. The benchmark results presented here may be useful for verifying further research and the present study suggests that the flexoelectric layer may be effectively exploited for advanced applications as a smart nano actuator.

A Symmetric Angle-Ply Composite Flywheel for High-Speed Energy Storage

This study was conducted to investigate the stress, strain, and strength ratio distributions in the composite flywheel rotor for high-energy density storage applications. Symmetric laminate design was used to avoid shear and extension–bending coupling and to minimize torsion coupling. The rotor studied consists of four anisotropic unidirectional plies. The continuity conditions of the radial stresses and displacements between plies were used to obtain a local stiffness matrix for each ply and develop the global stiffness matrix for the rotor due to the different ply orientations. The Tsai–Wu three-dimensional (3D) quadratic failure criterion in stress space was used to evaluate the strength ratio of the rings. Analysis was done for ply orientations between [±5 deg]S and [±85 deg]S. Three specific ply orientations were reported for discussion. The results show how the stress, strain, and safe rotational speed of the flywheel change as the ply orientations are varied. The circumferential stress was found to be the dominant stress. It increases as the ply angle increased in the circumferential direction while the axial stress decreased. Due to significant improvements in composite materials and technology, the results from this study will contribute to further development of the flywheel which has recently re-emerged as a promising application for energy storage.

Coupled instabilities of surface crease and bulk bending during fast free swelling of hydrogels.

Most studies on hydrogel swelling instability have been focused on a constrained boundary condition. In this paper, we studied the mechanical instability of a piece of disc-shaped hydrogel during free swelling. The fast swelling of the gel induces two swelling mismatches; a surface-inner layer mismatch and an annulus-disc mismatch, which lead to the formation of a surface crease pattern and a saddle-like bulk bending, respectively. For the first time, a stripe-like surface crease that is at a right angle on the two surfaces of the gel was observed. This stripe pattern is related to the mechanical coupling of surface instability and bulk bending, which is justified by investigating the swelling-induced surface pattern on thin hydrogel sheets fixed onto a saddle-shaped substrate prior to swelling. A theoretical mechanism based on an energy model was developed to show an anisotropic stripe-like surface crease pattern on a saddle-shaped surface. These results might be helpful to develop novel strategies for controlling crease patterns on soft and wet materials by changing their three-dimensional shape.