介电弹性体圆管的粘弹非线性动力学分析

曾修豪，黄志龙，王惠明; Zeng Xiuhao; Huang Zhilong; Wang Huiming

引 en

介电弹性体圆管的黏弹非线性动力学分析

曾修豪
机构：
浙江大学航空航天学院,杭州 310027
×
，黄志龙
机构：
浙江大学航空航天学院,杭州 310027
×
，王惠明
机构：
浙江大学航空航天学院,杭州 310027
×

浙江大学航空航天学院,杭州 310027；

Nonlinear Dynamic Analysis of Dielectric Elastomer Tube Considering the Viscoelastic Effect

Zeng Xiuhao
Affiliation：
School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027 ,China
×
，Huang Zhilong
Affiliation：
School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027 ,China
×
，Wang Huiming
Affiliation：
School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027 ,China
×

School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027 ,China；

通讯作者:

王惠明,E-mail∶wanghuiming@zju.edu.cn

中图分类号:O322;O343.5

文献标识码:A

文章编号:1672-6553-2023-21(2)-033-008

DOI:10.6052/1672-6553-2022-013

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参考文献
出版信息

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目录contents

摘要 Abstract
关键词 Keywords
1 理论建模
2 数值结果与讨论
2.1 内变量的演化规律
2.2 黏弹性对瞬时固有频率的影响
2.3 黏弹性对受迫振动响应的影响
3 结论
参考文献

摘要

研究了考虑黏弹性情形介电弹性体(DE)圆管的非线性动力学行为.利用弹簧-黏壶模型和不可压缩材料的neo-Hookean模型,建立了考虑黏弹性的介电弹性体圆管的动力学方程.采用小扰动方法,给出了瞬时固有频率的表达式,分析了特征时间、预拉伸以及圆管的厚度等参数对瞬时固有频率的影响规律.研究了周期电压作用下的非线性振动特性.结果表明黏弹性会显著降低响应振幅,但不会影响准静态最终固有频率.在没有直流电压时,系统仅产生单一主共振,但当直流电压足够大时,会发生多频共振.

Abstract

The nonlinear dynamic behavior of a dielectric elastomer (DE) tube considering the viscoelastic effect is investigated. By employing the spring-dashpot model and the neo-Hookean model of the incompressible soft materials, the governing equation is constructed for the viscoelastic DE tube. With the aid of the small disturbance method, the analytical expression of the instantaneous intrinsic frequency is obtained. The effect of the characteristic time, the prestretch and the thickness parameter on the instantaneous intrinsic frequency is studied. The nonlinear responses of the DE tube under the periodic voltage excitation are presented. It is shown that the amplitude of the response is greatly suppressed by the viscous effect, while the intrinsic frequency of the quasi-static state in the end is not affected by the viscous effect. Only the principal resonance is observed for the case that no DC voltage component is applied. The multiple frequency resonance phenomena are observed for the case that a large enough DC voltage component is applied.

关键词

介电弹性体；黏弹性；动力学特性；圆管；非线性

Keywords

dielectric elastomer ； viscoelastic effect ； dynamic behavior ； tube ； nonlinear

引言
介电弹性体（DE）因其有低密度、低模量、大驱动应变、响应迅速和高比能密度等特性在近二十年来受到广泛关注，应用领域拓展至仿生人工肌肉、软体机器人、可调焦透镜、触摸感应器和能量采集装置等^[1，2].
介电弹性体动态方面的研究已有很多报道，对于平面构型，Li等^[3]研究了机械载荷及电压作用下介电弹性膜的动力学特性，研究表明预拉伸及静电压可以调节结构的固有频率.Dubois等^[4]研究了电压对圆形膜谐振器共振频率的影响.Feng等^[5]研究了基于DE膜的微梁共振.Kollosche等^[6]研究了纯剪切DE作动器.盛俊杰等^[7]研究了交变电载荷作用下平面DE制动器的非线性动态特性.对于球形构型，Zhu等^[8]分析了DE球形膜的非线性振动.Wang^[9]建立了考虑空气耦合效应和环境压力及黏弹性的球形介电弹性体制动器的力学模型.Yong等^[10]和He等^[11]研究了厚壁介电弹性体球壳的非线性振动.Rudykh等^[12]研究了厚壁介电弹性体球壳的突跳.Liang等^[13]研究了内压和气压共同作用下介电弹性体球壳的分叉行为.金肖玲等^[14]研究了压力和电压作用下球膜的随机响应.对于柱壳构型，Jian等^[15]研究了介电弹性体圆管的机电失稳.Lu等^[16]分析了介电弹性体圆管的分叉和凸胀.Bortot^[17]研究了厚壁介电弹性体管在简谐或突加定常电载荷下的动力学响应.Son等^[18]给出了一种分析管状介电高弹体传感器动力学响应的理论分析模型.何新振等^[19]研究了介电弹性体柱壳在动态电压下的响应.王成敏等^[20]研究了周期载荷下介电弹性体圆柱壳的动力响应.
介电弹性体是一类聚合物材料，实验和理论研究表明，黏弹性效应对其力学性能具有重要影响^[21-27].从相关的文献可以看出，关于黏弹性效应对介电弹性体结构影响的研究主要集中在各种形状的薄膜结构，而对于黏弹性对中厚介电弹性体圆管动力学的影响，据作者所知，尚无文献开展相应研究.管状介电弹性体具有容易制备、密度低和价廉的特点，可用于大应变传感器^[18]、高性能制动器和浮标能量收集装置^[1].本文采用考虑黏弹性的neo-Hookean模型，利用Zhao等^[28]给出的内变量演化方程，建立黏弹性介电弹性体中厚圆管动力学问题的方程，进而研究黏弹性效应对介电弹性体中厚圆管动力学响应的影响规律，并研究了壁厚、电压等参数对瞬时固有频率和谐振特性的影响.
1 理论建模
图1所示为介电弹性体圆管，圆管的内外表面涂有电极.在参考状态，轴向长度为L，内外半径分别为A和B，经过轴向均匀预拉伸，轴向长度变为l，内外半径分别变为a和b.轴向伸长率为λ_z=l/L.横截面上半径为R的物质点移动至半径为r处，假设介电弹性体的体积不可压缩，可得

(r^{2} - a^{2}) l = (R^{2} - A^{2}) L

(1)

变形梯度可以写成

F = d i a g [\frac{1}{λ_{θ} λ_{z}}, λ_{θ}, λ_{z}]

(2)

式中，λ_θ=r/R.
图1 介电弹性体圆管的变形示意图
Fig.1 Schematic diagram of a dielectric elastomer tube coated with two compliant electrodes
图2 弹簧黏壶模型
Fig.2 Rheological model for a viscoelastic elastomer
考虑黏弹性效应，采用图2所示的流变模型^[28]，包含两个弹簧和一个黏壶.采用neo-Hookean模型，自由能函数可写为

\begin{matrix} W = \frac{μ_{α}}{2} (λ_{z}^{2} + λ_{θ}^{2} + λ_{z}^{- 2} λ_{θ}^{- 2} - 3) + \\ \frac{μ_{β}}{2} (λ_{z}^{2} ξ_{z}^{- 2} + λ_{θ}^{2} ξ_{θ}^{- 2} + λ_{z}^{- 2} λ_{θ}^{- 2} ξ_{z}^{2} ξ_{θ}^{2} - 3) + \\ \frac{{\tilde{D}}_{r}^{2}}{2 ε} λ_{z}^{- 2} λ_{θ}^{- 2} \end{matrix}

(3)

式中，μ_α和μ_β是切变模量，ξ_z和ξ_θ是内变量，表示黏壶的变形.ε是介电常数， ${\tilde{D}}_{}$ 是名义电位移，与真实电位移的关系是 ${\tilde{D}}_{r} = D_{r} λ_{z} λ_{θ}$ ，其中^[19]

D_{r} = \frac{Φ ε}{r l n (b / a)}

(4)

Φ为圆管内外表面电极上的电势差.根据图2所示的模型，可给出如下关系

λ_{z} = λ_{z}^{e} ξ_{z}, λ_{θ} = λ_{θ}^{e} ξ_{θ}

(5)

用η表示黏壶的黏度系数，引入松弛时间

τ = \frac{η}{μ_{β}}

(6)

内变量演化方程为^[28]

\begin{matrix} \frac{d ξ_{z}}{d t} = \frac{1}{τ} (λ_{z}^{2} ξ_{z}^{- 3} - λ_{z}^{- 2} λ_{θ}^{- 2} ξ_{z} ξ_{θ}^{2}) \\ \frac{d ξ_{θ}}{d t} = \frac{1}{τ} (λ_{θ}^{2} ξ_{θ}^{- 3} - λ_{z}^{- 2} λ_{θ}^{- 2} ξ_{z}^{2} ξ_{θ}) \end{matrix}

(7)

DE圆管的运动方程为

\frac{d σ_{r r}}{d r} + \frac{σ_{r r} - σ_{θ θ}}{r} = \ddot{ρ r}

(8)

环向和径向的真实应力差为

σ_{θ θ} - σ_{r r} = λ_{θ} \frac{\partial W}{\partial λ_{θ}}

(9)

由式（1）可导出

\ddot{r} = \frac{1}{r} [(1 - \frac{a^{2}}{r^{2}}) {\dot{a}}^{2} + a \ddot{a}]

(10)

将（8）式从内半径a积分到外半径b

\begin{matrix} \int_{a}^{b} \frac{d σ_{r r}}{d r} d r + \int_{a}^{b} \frac{1}{r} [μ_{α} λ_{r}^{2} - μ_{α} λ_{θ}^{2} + μ_{β} \\ (λ_{z}^{- 2} λ_{θ}^{- 2} ξ_{z}^{2} ξ_{θ}^{2} - λ_{θ}^{2} ξ_{θ}^{- 2}) + \frac{D_{r}^{2}}{ε}] d r = \int_{a}^{b} \ddot{ρ} d r \end{matrix}

(11)

引入变量替换

k = λ_{θ}, \frac{a}{A} = x, Δ = \frac{B^{2}}{A^{2}} - 1

(12)

且有 $k （ a ） = x ， k （ b ） = \sqrt{\frac{x^{2} + λ_{z}^{- 1} Δ}{Δ + 1}} .$
设内变量沿厚度方向变化不大，式（11）中与黏弹性有关的那项积分可化为

\begin{matrix} \int_{a}^{b} μ_{β} (λ_{z}^{- 2} λ_{θ}^{- 2} ξ_{z}^{2} ξ_{θ}^{2} - λ_{θ}^{2} ξ_{θ}^{- 2}) \frac{d r}{r} = \\ \frac{μ_{β}}{2} λ_{z}^{- 2} ξ_{z}^{2} ξ_{θ}^{2} (λ_{z} l n \frac{x^{2} + λ_{z}^{- 1} Δ}{x^{2}} - \frac{Δ + 1}{x^{2} + λ_{z}^{- 1} Δ} + x^{- 2}) - \\ \frac{μ_{β}}{2} λ_{z}^{- 1} ξ_{θ}^{- 2} l n (Δ + 1) \end{matrix}

(13)

由于体积不可压缩，公式（11）中的密度ρ为常数.式（11）的第一项即为作用在圆管外表面和内表面上的径向应力差，用σ₀表示.利用式（13）并将式（11）完成定积分，整理得到关于x的二阶非线性常微分方程

\ddot{x} + h {\dot{x}}^{2} + g = 0

(14)

其中

h = \frac{1}{x} [1 - \frac{Δ}{(λ_{z} x^{2} + Δ) γ_{1}}]

(15)

\begin{matrix} g = 2 A^{2} \frac{ρ}{μ_{α}} (- σ_{0} - f_{1} Φ^{2} - \frac{μ_{α}}{2} f_{2} - \\ \frac{μ_{β}}{2} f_{3}) / (ρ A^{2} x γ_{1}) \end{matrix}

(16)

\begin{matrix} f_{1} = \frac{ε Δ}{2 A^{2} x^{2} (λ_{z} x^{2} + Δ) γ_{1}^{2}} \\ f_{2} = λ_{z}^{- 1} l n \frac{x^{2} + λ_{z}^{- 1} Δ}{x^{2} (Δ + 1)} - λ_{z}^{- 2} γ_{2} + λ_{z}^{- 2} x^{- 2} \\ f_{3} = λ_{z}^{- 2} ξ_{z}^{2} ξ_{θ}^{2} (λ_{z} l n \frac{x^{2} + λ_{z}^{- 1} Δ}{x^{2}} - γ_{2} + \\ x^{- 2}) - λ_{z}^{- 1} ξ_{θ}^{- 2} l n (Δ + 1) \\ γ_{1} = l n (1 + \frac{Δ}{λ_{z} x^{2}}), γ_{2} = \frac{Δ + 1}{x^{2} + λ_{z}^{- 1} Δ} \end{matrix}

(17)

式（14）中已按 $t \to t / (A \sqrt{ρ / μ_{α}})$ 的方式对时间做了无量纲处理，后文不再更换时间符号t.式（7）和式（14）组成了考虑黏弹性影响的介电弹性体圆管非线性振动的控制方程.在模型中令μ_β=0且忽略演化方程（7），便退化为不考虑黏弹性影响的介电弹性体圆管的非线性振动^[19].需要指出的是，式（14）中关于 $h {\dot{x}}^{2}$ 的项实际上是由引入圆管的不可压缩条件导致的，它并非是非线性阻尼项，而是由于考虑不可压缩引起的附加惯性项，并且当厚度趋于零时该项退化为零.
2 数值结果与讨论
关于无黏性效应的动力响应已有详细研究^[19]，这里重点考虑黏弹性对系统动力特性的影响规律.首先讨论两个内变量ξ_z和ξ_θ的演化规律，然后讨论黏弹性对瞬时固有频率的影响，最后探讨周期电压作用下的动态响应.如没有特别说明，部分参数的取值为μ_α=μ_β=1MPa，A=0.01m，ρ=950kg/m³，σ₀=0，ε=2.2×10^-11F/m.
2.1 内变量的演化规律
由于伴随耗散过程，若载荷施加后保持恒定，圆管的振动将会逐渐停止，最终x会稳定到一个定值.图3给出了突加直流电压作用下介电弹性体圆管的黏弹性动力学解和静力学解的对比，由图3知，黏弹性动力学解最终趋于静力学解，该结果与物理现象相符.接下来讨论内变量在这个过程中的演化规律.
观察内变量演化方程（7），右边不显含时间项，是一个非线性自治系统.已知λ_z是常数，而λ_θ总会随着耗散而在一个恒定值x₀附近微幅振荡，不妨先令λ_θ=x₀，求得临界点（λ_z，x₀），在这一点上有两个相异的负特征值，所以它是稳定的结点.也就是说在这一点附近取任意初值，随着时间演化，内变量总会趋于这一点.图4是式（7）取λ_z=1.2，x₀=0.9时在临界点附近取不同初值后内变量的相图，可以看到无论取怎样的初值，内变量都会演化到临界点.
图3 黏弹性动力学解和静力学解的对比（Φ=20kV）
Fig.3 Comparison between the dynamic solution considering the viscoelastic effect and the static solution (Φ=20kV)
图5（a）是式（7）在给定t=0时刻的初值ξ_z=1和ξ_θ=1下内变量的时程.随着时间进程，在预拉伸（λ_z=1.2）情况下，z方向的内变量逐渐变大，最终趋于z方向的稳定值，θ方向内变量逐渐减小，最终趋于θ方向的稳定值.在预压缩（λ_z=0.8）情况下，z方向内变量逐渐减小，最终趋于z方向的稳定值，θ方向内变量逐渐变大，最终趋于θ方向的稳定值.
图4 不同初值下内变量的演化相图（λ_z=1.2）
Fig.4 The evolution of phase diagram of the internal variables under different initial values (λ_z=1.2)
由式（7）可知，ξ_z和ξ_θ与λ_z和λ_θ有关.接下来讨论内变量在厚度方向上的分布规律.由式（1）得

λ_{θ} (R, t) = \sqrt{(x^{2} - λ_{z}^{- 1}) (A / R)^{2} + λ_{z}^{- 1}}

(18)

给定初值，根据式（7）和方程（14）解得x（t），ξ_z（t）和ξ_θ（t）.将x（t）代入式（18）得到λ_θ（R，t），其中λ_z（R，t）=λ_z是不随时间和半径变化的常数，由式（7）可以解得ξ_z（R，t）和ξ_θ（R，t）.容易得知，z方向的内变量沿着厚度均匀分布.图5（b）给出了Δ=1时圆管在t=0，10，30，200这四个时刻θ方向内变量ξ_θ在半径上的分布曲线，ξ_θ在厚度方向变化不大，且越接近稳态，ξ_θ在厚度方向上的分布越均匀，因此式（13）积分时将内变量看成常数是合理的.
2.2 黏弹性对瞬时固有频率的影响
本节考察黏弹性对瞬时固有频率的影响.当圆管受到突加载荷作用时，在初始时刻，黏壶来不及变形，有ξ_z（0）=ξ_θ（0）=1，随着时间演化，黏壶逐渐发生变形，DE圆管的形状也发生变化.准静态解x_eq（t）可以由式（7）和式（19）求得

g (x_{e q}, F, ξ_{z}, ξ_{θ}) = 0

(19)

在演化过程的每个时刻t对x在准静态解x_eq（t）做扰动δ.δ和 $\dot{δ}$ 均是小量，略去它们的高阶项，可得线性化方程

\ddot{δ} + {\frac{\partial g}{\partial x}|}_{x = x_{e q} (t)} \cdot δ = 0

(20)

图5 内变量演化规律图
Fig.5 Evolution of the internal variables
在准静态x_eq（t）处的瞬时固有频率

ω = \sqrt{{\frac{\partial g}{\partial x}|}_{x = x_{e q} (t)}}

(21)

求解瞬时固有频率，需要联立式（21）和式（7）.求解过程简单总结为，在初始时刻给定ξ_z（0）=1和ξ_θ（0）=1，根据式（19）求得初始步的x_eq₀，利用方程（7）求得下一步的内变量ξ_z（Δt），ξ_θ（Δt）.循环利用方程（19）和方程（7），可得任一时刻的x_eq（t）及ξ_z（t）和ξ_θ（t），最后将x_eq（t）代入式（21）求得固有频率的演化规律.
图6是使用变步长的欧拉法求得的三个特征时间τ=20s，200s和800s对应的瞬时固有频率演化曲线.瞬时固有频率的变化规律均是先降低后增加，最后逐渐趋于一个定值.特征时间的不同不影响最终的固有频率.
图6 不同特征时间的瞬时固有频率演化
Fig.6 Evolution of the instantaneous intrinsic frequency for different characteristic times
图7 不同系统参数对瞬时固有频率的影响
Fig.7 Influence of system parameters on the instantaneous intrinsic frequency
图8 正弦电压作用下的位移响应和相图
Fig.8 Displacement response and phase diagram under sinusoidal voltage
由式（12）的第三式可知，Δ是描述DE圆管厚度的参量，其值大于0，Δ值越大表示圆管越厚.图7（a）给出了外载荷相同情形下不同厚度圆管固有频率的演化曲线.结果表明，log（1+t）（t为无量纲时间）在[2，5]的区间是瞬时固有频率变化的敏感区间，且Δ越大，也就是圆管越厚，瞬时固有频率越低.有趣的是，瞬时固有频率的变化并不是单调的，是先减小后增大，最后稳定下来，这是由于轴向、环向和径向的伸长率和内变量等共同作用的结果.图7（b）给出了经预拉伸（或预压缩）情况下瞬时固有频率的演化规律，当z向预拉伸（λ_z＞1）时，随着时间演化，瞬时固有频率逐渐增大，且预拉伸值越大，演化到最终稳定状态的瞬时固有频率越大，而当z向预压缩（λ_z＜1）时，瞬时固有频率逐渐减小，且预压缩值越大，演化到最终稳定状态的瞬时固有频率越小.
2.3 黏弹性对受迫振动响应的影响
为研究黏弹性对DE圆管受迫振动的影响，假设圆管受到如下的时变电压载荷

Φ (t) = Φ_{d c} + Φ_{a c} s i n (Ω t)

(22)

图9 不同特征时间周期性电压作用下的频响曲线
Fig.9 The frequency response curve under periodic voltage for different characteristic times
其中，Φ_dc是直流电压分量，Φ_ac是交流分量的幅值，Ω是交流电压的角频率.联立求解方程（7）和方程（14），可得考虑黏弹性效应的DE圆管的动态响应.图8（a）是在两种不同频率的周期载荷作用下的位移响应时程，图8（b）是相应周期运动的相图，结果表明，圆管在远离特征时间以后表现出了周期运动.
当系统受到周期激励，定义远离特征时间τ以后一段时间内位移的最大值和最小值差值的一半为周期解的幅值A.图9（a）和图9（b）分别是有和无直流电压分量时频率在0.1~5之间的频响曲线，当电压的频率接近准静态最终固有频率时，周期解的幅值显著提高.同样可以看到在三种特征时间下的准静态最终固有频率几乎不变，但随着特征时间减小，共振幅值也会显著降低.这是由于特征时间的减小，相当于系统的黏性效应更加明显，从而使共振幅值显著降低.由图9可知，当没有直流电压分量时，由于式（16）中交变电压是Φ²，其实际激励频率为2Ω，因此频响曲线图9（b）只有一个共振峰，而存在较大的直流电压时，交变电压Φ²存在激励频率分别为Ω及2Ω的谐和激励，因此频响曲线图9（a）出现两个共振峰.由图7~9可知，瞬时固有频率的准静态最终固有频率、频响曲线的主共振频率和时程曲线的频率三者的结果一致，从另一方面表明数值结果的可信性.
3 结论
本研究建立了考虑黏弹性效应的介电弹性体圆管的动力学方程，研究表明，黏弹性对介电弹性体圆管的瞬时固有频率有显著的影响，特征时间影响瞬时固有频率的变化敏感区间，但不影响最终瞬时固有频率的大小.轴向预拉伸使得瞬时固有频率升高，而轴向预压缩使得瞬时固有频率降低.壁厚的增加使得瞬时固有频率降低.特征时间对周期性电压作用下DE圆管的响应也有显著影响，特征时间越短，相当于黏性效应越强，共振幅值越小，当直流电压足够大时，会发生多频主共振，而当没有直流电压时，只发生单一主共振.
参考文献
- [1] LU T,MA C,WANG T.Mechanics of dielectric elastomer structures∶ A review [J].Extreme Mechanics Letters,2020,38∶ 100752.
- [2] MORETTI G,ROSSET S,VERTECHY R,et al.A review of dielectric elastomer generator systems [J].Advanced Intelligent Systems,2020,2(10)∶ 2000125.
- [3] LI T F,QU S X,YANG W.Electromechanical and dynamic analyses of tunable dielectric elastomer resonator [J].International Journal of Solids and Structures,2012,49(26)∶ 3754-3761.
- [4] DUBOIS P,ROSSET S,NIKLAUS M,et al.Voltage Control of the resonance frequency of dielectric electroactive polymer(DEAP)membranes [J].Journal of Microelectromechanical Systems,2008,17(5)∶1072-1081.
- [5] FENG C,JIANG L,LAU W M.Dynamic characteristics of a dielectric elastomer-based microbeam resonator with small vibration amplitude [J].Journal of Micromechanics and Microengineering,2011,21(9)∶ 095002.
- [6] KOLLOSCHE M,ZHU J,SUO Z,et al.Complex interplay of nonlinear processes in dielectric elastomers [J].Physical Review E,2012,85(5)∶ 051801.
- [7] 盛俊杰,李树勇,张玉庆,等.介电弹性体材料致动器的非线性动态行为研究 [J].动力学与控制学报,2017,15(2)∶ 119-124.SHENG J J,LI S Y,ZHANG Y Q,et al.Nonlinear dynamic performance of a dielectric elastomer actuator [J].Journal of Dynamics and Control,2017,15(2)∶ 119-124.(in Chinese)
- [8] ZHU J,CAI S,SUO Z.Nonlinear oscillation of a dielectric elastomer balloon [J].Polymer International,2010,59(3)∶378-383.
- [9] WANG H M.Temporal evolution in a dissipative air-coupled spherical dielectric elastomer actuator [J].Journal of Mechanical Science and Technology,2017,31(9)∶ 4337-4343.
- [10] YONG H,HE X,ZHOU Y.Dynamics of a thick-walled dielectric elastomer spherical shell [J].International Journal of Engineering Science,2011,49(8)∶ 792-800.
- [11] HE X,YONG H,ZHOU Y.The characteristics and stability of a dielectric elastomer spherical shell with a thick wall [J].Smart Materials and Structures,2011,20(5)∶ 055016.
- [12] RUDYKH S,BHATTACHARYA K,DEBOTTON G.Snap-through actuation of thick-wall electroactive balloons [J].International Journal of Non-Linear Mechanics,2012,47(2)∶ 206-209.
- [13] LIANG X,CAI S.Shape bifurcation of a spherical dielectric elastomer balloon under the actions of internal pressure and electric voltage [J].Journal of Applied Mechanics,2015,82(10)∶ 101002.
- [14] 金肖玲,王永,黄志龙.气压扰动下介电弹性体球膜的随机响应分析 [J].动力学与控制学报,2017,15(3)∶ 250-255.JIN X L,WANG Y,HUANG Z L.Random response of dielectric elastomer balloon subjected to disturbed pressure [J].Journal of Dynamics and Control,2017,15(3)∶ 250-255.(in Chinese)
- [15] JIAN Z,STOYANOV H,KOFOD G,et al.Large deformation and electromechanical instability of a dielectric elastomer tube actuator [J].Journal of Applied Physics,2010,108(7)∶ 074113.
- [16] LU T Q,AN L,LI J G,et al.Electro-mechanical coupling bifurcation and bulging propagation in a cylindrical dielectric elastomer tube [J].Journal of the Mechanics and Physics of Solids,2015,85∶ 160-175.
- [17] BORTOT E.Nonlinear dynamic response of soft thick-walled electro-active tubes [J].Smart Materials and Structures,2018,27∶ 105025.
- [18] SON S,GOULBOURNE N C.Dynamic response of tubular dielectric elastomer transducers [J].International Journal of Solids and Structures,2010,47(20)∶ 2672-2679.
- [19] 何新振,雍华东,周又和.电活性聚合物圆柱壳静态与动态电压下的响应及稳定性 [J].固体力学学报,2012,33(4)∶ 341-348.HE X Z,YONG H D,ZHOU Y H.The dynamic response and stability of a dielectric elastomer cylindrical shell under static and periodic votage [J].Acta Mechanica Solida Sinica,2012,33(4)∶ 341-348.(in Chinese)
- [20] 王成敏,任九生.周期载荷下电活性聚合物圆柱壳的动力响应 [J].动力学与控制学报,2016,14(3)∶ 211-216.WANG C M,REN J S.Dynamical response of electro-active polymer cylindrical shells under periodic pressure [J].Journal of Dynamics and Control,2016,14(3)∶ 211-216.(in Chinese)
- [21] FOO C C,CAI S,KOH S,et al.Model of dissipative dielectric elastomers [J].Journal of Applied Physics,2012,111(3)∶ 836-382.
- [22] WANG H M,LEI M,CAI S Q.Viscoelastic deformation of a dielectric elastomer membrane subject to electromechanical loads [J].Journal of Applied Physics,2013,113∶ 213508.
- [23] KOLLOSCHE M,KOFOD G,SUO Z,at al.Temporal evolution and instability in a viscoelastic dielectric elastomer [J].Journal of the Mechanics and Physics of Solids,2015,76∶ 47-64.
- [24] LIU F,ZHOU J X.Shooting and arc-length continuation method for periodic solution and bifurcation of nonlinear oscillation of viscoelastic dielectric elastomers [J].ASME Journal of Applied Mechanics 2018,85∶ 011005.
- [25] LI Y,OH I,CHEN J,et al.Nonlinear dynamic analysis and active control of visco-hyperelastic dielectric elastomer membrane [J].International Journal of Solids and Structures,2018,152-153∶ 28-38.
- [26] KASHYAP K,SHARMA A K,JOGLEKAR M M.Nonlinear dynamic analysis of aniso-viscohyperelastic dielectric elastomer actuators [J].Smart Materials and Structures,2020,29∶ 055014.
- [27] LI J,SUN L J,NGUYEN T D,et al.Research and analysis of resonant and stiffness of cylindrical dielectric elastomer actuator [J].Materials Research Express,2021,8∶ 065701.
- [28] ZHAO X,KOH S,SUO Z.Nonequilibrium thermodynamics of dielectric elastomers [J].International Journal of Applied Mechanics,2011,3(2)∶ 203-217.

基本信息

中图分类号: O322;O343.5
文献标识码: A
DOI: 10.6052/1672-6553-2022-013
文章编号: 1672-6553-2023-21(2)-033-008

基金信息

引用信息

稿件历史

收稿日期: 2022-01-28

参考文献

[1] LU T,MA C,WANG T.Mechanics of dielectric elastomer structures∶ A review [J].Extreme Mechanics Letters,2020,38∶ 100752.
[2] MORETTI G,ROSSET S,VERTECHY R,et al.A review of dielectric elastomer generator systems [J].Advanced Intelligent Systems,2020,2(10)∶ 2000125.
[3] LI T F,QU S X,YANG W.Electromechanical and dynamic analyses of tunable dielectric elastomer resonator [J].International Journal of Solids and Structures,2012,49(26)∶ 3754-3761.
[4] DUBOIS P,ROSSET S,NIKLAUS M,et al.Voltage Control of the resonance frequency of dielectric electroactive polymer(DEAP)membranes [J].Journal of Microelectromechanical Systems,2008,17(5)∶1072-1081.
[5] FENG C,JIANG L,LAU W M.Dynamic characteristics of a dielectric elastomer-based microbeam resonator with small vibration amplitude [J].Journal of Micromechanics and Microengineering,2011,21(9)∶ 095002.
[6] KOLLOSCHE M,ZHU J,SUO Z,et al.Complex interplay of nonlinear processes in dielectric elastomers [J].Physical Review E,2012,85(5)∶ 051801.
[7] 盛俊杰,李树勇,张玉庆,等.介电弹性体材料致动器的非线性动态行为研究 [J].动力学与控制学报,2017,15(2)∶ 119-124.SHENG J J,LI S Y,ZHANG Y Q,et al.Nonlinear dynamic performance of a dielectric elastomer actuator [J].Journal of Dynamics and Control,2017,15(2)∶ 119-124.(in Chinese)
[8] ZHU J,CAI S,SUO Z.Nonlinear oscillation of a dielectric elastomer balloon [J].Polymer International,2010,59(3)∶378-383.
[9] WANG H M.Temporal evolution in a dissipative air-coupled spherical dielectric elastomer actuator [J].Journal of Mechanical Science and Technology,2017,31(9)∶ 4337-4343.
[10] YONG H,HE X,ZHOU Y.Dynamics of a thick-walled dielectric elastomer spherical shell [J].International Journal of Engineering Science,2011,49(8)∶ 792-800.
[11] HE X,YONG H,ZHOU Y.The characteristics and stability of a dielectric elastomer spherical shell with a thick wall [J].Smart Materials and Structures,2011,20(5)∶ 055016.
[12] RUDYKH S,BHATTACHARYA K,DEBOTTON G.Snap-through actuation of thick-wall electroactive balloons [J].International Journal of Non-Linear Mechanics,2012,47(2)∶ 206-209.
[13] LIANG X,CAI S.Shape bifurcation of a spherical dielectric elastomer balloon under the actions of internal pressure and electric voltage [J].Journal of Applied Mechanics,2015,82(10)∶ 101002.
[14] 金肖玲,王永,黄志龙.气压扰动下介电弹性体球膜的随机响应分析 [J].动力学与控制学报,2017,15(3)∶ 250-255.JIN X L,WANG Y,HUANG Z L.Random response of dielectric elastomer balloon subjected to disturbed pressure [J].Journal of Dynamics and Control,2017,15(3)∶ 250-255.(in Chinese)
[15] JIAN Z,STOYANOV H,KOFOD G,et al.Large deformation and electromechanical instability of a dielectric elastomer tube actuator [J].Journal of Applied Physics,2010,108(7)∶ 074113.
[16] LU T Q,AN L,LI J G,et al.Electro-mechanical coupling bifurcation and bulging propagation in a cylindrical dielectric elastomer tube [J].Journal of the Mechanics and Physics of Solids,2015,85∶ 160-175.
[17] BORTOT E.Nonlinear dynamic response of soft thick-walled electro-active tubes [J].Smart Materials and Structures,2018,27∶ 105025.
[18] SON S,GOULBOURNE N C.Dynamic response of tubular dielectric elastomer transducers [J].International Journal of Solids and Structures,2010,47(20)∶ 2672-2679.
[19] 何新振,雍华东,周又和.电活性聚合物圆柱壳静态与动态电压下的响应及稳定性 [J].固体力学学报,2012,33(4)∶ 341-348.HE X Z,YONG H D,ZHOU Y H.The dynamic response and stability of a dielectric elastomer cylindrical shell under static and periodic votage [J].Acta Mechanica Solida Sinica,2012,33(4)∶ 341-348.(in Chinese)
[20] 王成敏,任九生.周期载荷下电活性聚合物圆柱壳的动力响应 [J].动力学与控制学报,2016,14(3)∶ 211-216.WANG C M,REN J S.Dynamical response of electro-active polymer cylindrical shells under periodic pressure [J].Journal of Dynamics and Control,2016,14(3)∶ 211-216.(in Chinese)
[21] FOO C C,CAI S,KOH S,et al.Model of dissipative dielectric elastomers [J].Journal of Applied Physics,2012,111(3)∶ 836-382.
[22] WANG H M,LEI M,CAI S Q.Viscoelastic deformation of a dielectric elastomer membrane subject to electromechanical loads [J].Journal of Applied Physics,2013,113∶ 213508.
[23] KOLLOSCHE M,KOFOD G,SUO Z,at al.Temporal evolution and instability in a viscoelastic dielectric elastomer [J].Journal of the Mechanics and Physics of Solids,2015,76∶ 47-64.
[24] LIU F,ZHOU J X.Shooting and arc-length continuation method for periodic solution and bifurcation of nonlinear oscillation of viscoelastic dielectric elastomers [J].ASME Journal of Applied Mechanics 2018,85∶ 011005.
[25] LI Y,OH I,CHEN J,et al.Nonlinear dynamic analysis and active control of visco-hyperelastic dielectric elastomer membrane [J].International Journal of Solids and Structures,2018,152-153∶ 28-38.
[26] KASHYAP K,SHARMA A K,JOGLEKAR M M.Nonlinear dynamic analysis of aniso-viscohyperelastic dielectric elastomer actuators [J].Smart Materials and Structures,2020,29∶ 055014.
[27] LI J,SUN L J,NGUYEN T D,et al.Research and analysis of resonant and stiffness of cylindrical dielectric elastomer actuator [J].Materials Research Express,2021,8∶ 065701.
[28] ZHAO X,KOH S,SUO Z.Nonequilibrium thermodynamics of dielectric elastomers [J].International Journal of Applied Mechanics,2011,3(2)∶ 203-217.

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介电弹性体圆管的黏弹非线性动力学分析

Nonlinear Dynamic Analysis of Dielectric Elastomer Tube Considering the Viscoelastic Effect

摘要

Abstract

关键词

Keywords

1 理论建模

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

2 数值结果与讨论

2.1 内变量的演化规律

(18)

2.2 黏弹性对瞬时固有频率的影响

(19)

(20)

(21)

2.3 黏弹性对受迫振动响应的影响

(22)

3 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

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