Nonlinear dynamic analysis of dielectric elastomer membrane with electrostriction

The dielectric elastomer (DE) is an important intelligent soft material widely used in soft actuators, and the dynamic response of the DE is highly nonlinear due to the material properties. In the DE, electrostriction denotes the deformation-dependent permittivity. In the present study, we formulate the nonlinear dynamic governing equations of the DE membrane considering the electrostriction effect. The free vibration and parametric excitation of the DE membrane with different geometric sizes are calculated. The free vibration bifurcations induced by the initial location and the voltage are both discussed according to an energy-based approach. The amplitude-frequency characteristics and bifurcation diagrams of parametric excitation are also given. The results show that electrostriction decreases the free vibration amplitude and increases the frequency, but it has less influence on the parametric excitation oscillation frequency and decreases the parametric excitation amplitude except when the membrane resonates. The initial location and the applied voltage can induce the snap-through instability of the free vibration. A large geometric size will lead to a much lower resonance frequency. The resonance amplitudes increase while the resonance frequencies decrease with the increase in the applied voltage. The critical voltage of snap-through instability for the parametric excitation is larger than that for the free vibration one.

     

Dynamic stability and bifurcation analysis in fractional thermodynamics

In mechanics, viscoelasticity was the first field of applications in studying geomaterials. Further possibilities arise in spatial non-locality. Non-local materials were already studied in the 1960s by several authors as a part of continuum mechanics and are still in focus of interest because of the rising importance of materials with internal micro- and nano-structure. When material instability gained more interest, non-local behavior appeared in a different aspect. The problem was concerned to numerical analysis, because then instability zones exhibited singular properties for local constitutive equations. In dynamic stability analysis, mathematical aspects of non-locality were studied by using the theory of dynamic systems. There the basic set of equations describing the behavior of continua was transformed to an abstract dynamic system consisting of differential operators acting on the perturbation field variables. Such functions should satisfy homogeneous boundary conditions and act as indicators of stability of a selected state of the body under consideration. Dynamic systems approach results in conditions for cases, when the differential operators have critical eigenvalues of zero real parts (dynamic stability or instability conditions). When the critical eigenvalues have non-trivial eigenspace, the way of loss of stability is classified as a typical (or generic) bifurcation. Our experiences show that material non-locality and the generic nature of bifurcation at instability are connected, and the basic functions of the non-trivial eigenspace can be used to determine internal length quantities of non-local mechanics. Fractional calculus is already successfully used in thermo-elasticity. In the paper, non-locality is introduced via fractional strain into the constitutive relations of various conventional types. Then, by defining dynamic systems, stability and bifurcation are studied for states of thermo-mechanical solids. Stability conditions and genericity conditions are presented for constitutive relations under consideration.     

Material instabilities of anisotropic saturated multiphase porous media

This paper analyses the material instability of fully saturated multiphase porous media. On account of the fact that anisotropic mechanical behaviours are widely observed in saturated and partially saturated geomaterials, the anisotropic constitutive model developed by Rudnicki for geomaterials is used to model the anisotropic mechanical behaviour of the solid skeleton of saturated porous geomaterials in axisymmetric compression test. The inertial coupling effect between solid skeleton and pore fluid is also taken into account in dynamic cases. Conditions for static instability (strain localisation) and dynamic instability (stationary discontinuity and flutter instability) of fully saturated porous media are derived. The critical modulus, shear band angle for strain localisation, and the bound within which flutter instability may occur are given in explicit forms. The effects of material parameters on material instability are investigated in detail by numerical computations.     

Dynamic non-linear behavior and stability of a ventricular assist device

This paper investigates the non-linear dynamic behavior and stability of the internal membrane of a ventricular assist device (VAD). This membrane separates the blood chamber from the pneumatic chamber, transmitting the driving cyclic pneumatic loading to blood flowing from the left ventricle into the aorta. The membrane is a thin, nearly spherical axi-symmetric shallow cap made of polyurethane and reinforced with a cotton mesh. Experimental evidence shows that the reinforced membrane behaves as an isotropic elastic material and exhibits both membrane and flexural stiffness. So, the membrane is modeled as an isotropic pressure loaded shallow spherical shell and its dynamic behavior and snap-through buckling considering different types of dynamic excitation relevant to the understanding of the VAD behavior is investigated. Based on Marguerre kinematical assumptions, the governing partial differential equations of motion are presented in the form of a compatibility equation and a transverse motion equation. The results show that the shell, when subjected to compressive pressure loading, may loose its stability at a limit point, jumping to an inverted position. If the compressive load is removed, the shell jumps back to its original configuration. This non-linear behavior is the key feature in the VAD behavior.     

旋转薄圆盘的行波动力学特性分析

旋转圆盘是广泛应用于旋转机械装置中的基本结构元件,圆盘在高速旋转状态下会表现出与低速或非旋转状态下迥异的力学性能.本文对高速旋转薄圆盘横向振动的行波动力学特性进行了分析,建立了考虑离心力引起的薄膜内力影响下的动力学控制方程以及相应的边界条件.采用伽辽金法数值模拟了旋转圆盘前、后行波振动频率和动力屈曲失稳临界转速随着圆盘几何参数如半径比、厚度的变化规律,以及材料参数对于振动频率和临界转速的影响等.本文的数值计算可以同时给出圆盘旋转的前、后行波频率,并且结果与实验结果吻合良好.     

Influence of system parameters on dynamic behavior of gear pair with stochastic backlash

The influence of stochastic backlash on nonlinear dynamic behavior of spur gear pair with stochastic assembling backlash is discussed. Bifurcation diagram and maximum Lyapunov exponent diagram of the system are presented to evaluate the influences of load ratio, damping ratio, and backlash on the dynamic behavior of the system. The results show that backlash has great contribution to light-loaded gear pair. Therefore, dynamic behavior with stochastic assembling backlash is analyzed. Two novel indexes are introduced to evaluate the dynamic behavior of the system, dynamic instability exponent and critical variance. In addition, the relationship between the dynamic instability exponent and variance of the backlash, and the relationship between the critical variance and the mean value of the backlash are studied. The results show that it is not the unique way to improve the dynamic behavior of the system by minimizing the assembling backlash, which provide a novel way to determine the machining precision requirements, tolerance for assembling, and etc.     

Material and structural instabilities of single-wall carbon nanotubes

The nonlinear atomistic interactions usually involve softening behavior. Instability resulting directly from this softening are called the material instability, while those unrelated to this softening are called the structural instability. We use the finite-deformation shell theory based on the interatomic potential to show that the tension instability of single-wall carbon nanotubes is the material instability, while the compression and torsion instabilities are structural instability.     

THEORY OF DIELECTRIC ELASTOMERS

In response to a stimulus, a soft material deforms, and the deformation provides a function. We call such a material a soft active material （SAM）. This review focuses on one class of soft active materials： dielectric elastomers. When a membrane of a dielectric elastomer is subject to a voltage through its thickness, the membrane reduces thickness and expands area, possibly straining over 100%. The dielectric elastomers are being developed as transducers for broad applications, including soft robots, adaptive optics, Braille displays, and electric generators. This paper reviews the theory of dielectric elastomers, developed within continuum mechanics and thermodynamics, and motivated by molecular pictures and empirical observations. The theory couples large deformation and electric potential, and describes nonlinear and nonequilibrium behavior, such as electromechanical instability and viscoelasticity. The theory enables the finite element method to simulate transducers of realistic configurations, predicts the efficiency of electromechanical energy conversion, and suggests alternative routes to achieve giant voltage-induced deformation. It is hoped that the theory will aid in the creation of materials and devices.     

The effect of plastic anisotropy on compressive instability in sheet metal forming

The wrinkling behavior of a thin sheet with perfect geometry is associated with compressive instability. The compressive instability is influenced by many factors such as stress state, mechanical properties of the sheet material, geometry of the body, contact conditions and plastic anisotropy. The analysis of compressive instability in a plastically deforming body is difficult considering all the factors because the effects of the factors are very complex and the instability behavior may show a wide variation for a small deviation of the factors. In this study, the bifurcation theory is introduced for the finite element analysis of puckering initiation and growth of a thin sheet with perfect geometry. All the above mentioned factors are conveniently considered by the finite-element method. The instability limit is found by the incremental analysis and the post-bifurcation behavior is analyzed by introducing the branching scheme proposed by Riks. The finite-element formulation is based on the incremental deformation theory and elastic–plastic material modeling. The finite-element analysis is carried out using the continuum-based resultant shell elements considering the anisotropy of the sheet metal. In order to investigate the effect of plastic anisotropy on the compressive instability, a square plate that is subjected to compression in one direction and tension in the other direction is analyzed by the above-mentioned finite-element analysis. The critical stress ratios above which buckling does not take place are found for various plastic anisotropic modeling methods and discussed. Finally, the effect of plastic anisotropy on the puckering behavior in the spherical cup deep drawing process is investigated. From the results of the finite-element analysis, it is shown that puckering behavior of sheet metal is largely affected by plastic anisotropy.     

The oscillatory behavior, static and dynamic analyses of a micro/nano gyroscope considering geometric nonlinearities and intermolecular forces

The nonlinear dynamic and static deflection of a micro/nano gyroscope under DC voltages and base rotation are investigated. The gyroscope undertakes two cou- pled bending motions along the drive and sense directions and subjected to electrostatic actuations and intermolecular forces. The nonlinear governing equations of motion for the system with the effect of electrostatic force, intermolecular tractions and base rotation are derived using extended Hamilton principle. Under constant voltage, the gyroscope finds the preformed shape. First, the deflection of the rnicro/nano gyroscope under electrostatic forces is obtained by static and dynamic analyses. Furthermore, the static and dynamic in- stability of the system are investigated. Afterward the oscillatory behavior of the pre-deformed micro/nano gyroscope around equilibrium is studied. The effects of intermolecular and nonlinear parameters on the static and dynamic de- flection, natural frequencies and instability of the micro/nano gyroscope are studied. The presented model can be used to exactly determine static and the dynamic behavior of vibratory micro/nano gyroscopes.     

直升机旋翼/机体动稳定性研究进展

首先对直升机旋翼/机体动不稳定性问题的种类进行了简要概述,包括旋翼挥舞/变距、变距/摆振、挥舞/摆振和挥舞/摆振/变距耦合等孤立旋翼动不稳定性问题,以及直升机地面共振和空中共振等旋翼/机体耦合动不稳定性问题,然后分别从气动力与结构的高精度数值模型、动稳定性的计算分析方法和实验模型测试3 个方面详细介绍了直升机旋翼/机体动不稳定性问题的研究现状,并着重讨论了直升机旋翼/机体动稳定性分析技术最近的主要研究方向:耦合CFD(computational fluid dynamics)/CSD(computational structuraldynamics) 的直升机旋翼气弹动稳定性分析、复合材料旋翼动稳定性分析及其材料不确定性影响、带减摆器的旋翼/机体动稳定性分析和先进直升机构型的旋翼/机体动稳定性分析,最后对直升机旋翼/机体动稳定性研究的发展趋势进行了展望.     

Temporal evolution and instability in a viscoelastic dielectric elastomer

Dielectric elastomer transducers are being developed for applications in stretchable electronics, tunable optics, biomedical devices, and soft machines. These transducers exhibit highly nonlinear electromechanical behavior: a dielectric membrane under voltage can form wrinkles, undergo snap-through instability, and suffer electrical breakdown. We investigate temporal evolution and instability by conducting a large set of experiments under various prestretches and loading rates, and by developing a model that allows viscoelastic instability. We use the model to classify types of instability, and map the experimental observations according to prestretches and loading rates. The model describes the entire set of experimental observations. A new type of instability is discovered, which we call wrinkle-to-wrinkle transition. A flat membrane at a critical voltage forms wrinkles and then, at a second critical voltage, snaps into another state of winkles of a shorter wavelength. This study demonstrates that viscoelasticity is essential to the understanding of temporal evolution and instability of dielectric elastomers.     

Structural bifurcation for ductile necking localization

Necking localization is common unstable behavior in ductile solids. This paper describes the unified necking localization mechanism. After describing one-dimensional instability problem, general material and structural instability criteria are formulated and the formulation is validated by non-linear finite element analysis. The trigger of necking localization is structural bifurcation and the behavior from a uniformly deformed state to ultimate localization just before fracture is continuous structural instability.     

Stability of parametric vibrations of hybrid composite plates

In this paper, the dynamic stability of laminated hybrid composite plates subjected to periodic uniaxial stress and bending stress is studied. The governing equations of motion of Mathieu-type are established by using the Galerkin method with reduced eigenfunctions transforms. Based on Bolotin's method the regions of dynamic instability of laminated hybrid composite plates are determined by solving the eigenvalue problems. The effects of layer thickness ratio, layer number, core material and load parameter on the dynamic instability of laminated hybrid composite plates are investigated and discussed.     

Instability analysis and simulation of water infiltration into an unsaturated elasto-viscoplastic material

A linear instability analysis was performed in order to investigate which variables have a significant effect on the onset of the instability of an unsaturated viscoplastic material subjected to water infiltration. It was found that the onset of the growing instability of the material system mainly depends on the specific moisture capacity, the suction, and the hardening parameter. Then, in order to simulate the water infiltration process of a one-dimensional unsaturated soil column, a multiphase coupled elasto-viscoplastic finite element analysis was performed based on the theory of porous media. The results of the numerical simulations are discussed with respect to the effect of the specific moisture capacity and the initial suction on the development of volumetric strain. We found that rapid transitions from unsaturated to saturated states and higher levels of initial suction lead to the contractive behavior of the material and instability. The instability detected by the numerical results is consistent with the theoretical results obtained through the linear instability analysis.     

An aeroelastic model for composite rotor blades with straight and swept tips. Part I: Aeroelastic stability in hover

An analytical model for predicting the aeroelastic behavior of composite rotor blades with straight and swept tips is presented. The blade is modeled by beam type finite elements along the elastic axis. A single finite element is used to model the swept tip. The non-linear equations of motion for the finite element model are derived using Hamilton's principle and based on a moderate deflection theory and accounts for: arbitrary cross-sectional shape, pretwist, generally anisotropic material behavior, transverse shears and out-of-plane warping. Numerical results illustrating the effects of tip sweep, anhedral and composite ply orientation on blade aeroelastic behavior are presented. It is shown that composite ply orientation has a substantial effect on blade stability. At low thrust conditions, certain ply orientations can cause instability in the lag mode. The flap-torsion coupling associated with tip sweep can also induce aeroelastic instability in the blade. This instability can be removed by appropriate ply orientation in the composite construction.     

Thermally induced dynamic instability of laminated composite conical shells

Thermally induced dynamic instability of laminated composite conical shells is investigated by means of a perturbation method. The laminated composite conical shells are subjected to static and periodic thermal loads. The linear instability approach is adopted in the present study. A set of initial membrane stresses due to the elevated temperature field is assumed to exist just before the instability occurs. The formulation begins with three-dimensional equations of motion in terms of incremental stresses perturbed from the state of neutral equilibrium. After proper nondimensionalization, asymptotic expansion and successive integration, we obtain recursive sets of differential equations at various levels. The method of multiple scales is used to eliminate the secular terms and make an asymptotic expansion feasible. Using the method of differential quadrature and Bolotin's method, and imposing the orthonormality and solvability conditions on the present asymptotic formulation, we determine the boundary frequencies of dynamic instability regions for various orders in a consistent and hierarchical manner. The principal instability regions of cross-ply conical shells with simply supported–simply supported boundary conditions are studied to demonstrate the performance of the present asymptotic theory.