A phenomenological constitutive model for the nonlinear viscoelastic responses of biodegradable polymers

We formulate a constitutive framework for biodegradable polymers that accounts for nonlinear viscous behavior under regimes with large deformation. The generalized Maxwell model is used to represent the degraded viscoelastic response of a polymer. The large-deformation, time-dependent behavior of viscoelastic solids is described using an Ogden-type hyperviscoelastic model. A deformation-induced degradation mechanism is assumed in which a scalar field depicts the local state of the degradation, which is responsible for the changes in the material’s properties. The degradation process introduces another timescale (the intrinsic material clock) and an entropy production mechanism. Examples of the degradation of a polymer under various loading conditions, including creep, relaxation and cyclic loading, are presented. Results from parametric studies to determine the effects of various parameters on the process of degradation are reported. Finally, degradation of an annular cylinder subjected to pressure is also presented to mimic the effects of viscoelastic arterial walls (the outer cylinder) on the degradation response of a biodegradable stent (the inner cylinder). A general contact analysis is performed. As the stiffness of the biodegradable stent decreases, stress reduction in the stented viscoelastic arterial wall is observed. The integration of the proposed constitutive model with finite element software could help a designer to predict the time-dependent response of a biodegradable stent exhibiting finite deformation and under complex mechanical loading conditions.     

Random response of a viscoelastic dielectric elastomer spherical balloon

The viscoelastic behavior of the dielectric elastomers (DEs) plays an important role in the time-dependent mechanical deformation. The increasingly demands to precise utilities and controls of the DEs are motivating the researchers to insight into the influence of the viscoelasticity on the dynamical response. In this paper, the random response of a DE spherical balloon subjected to electrical and/or mechanical disturbances is analytically studied. Based on a rheological model of two parallel units, the governing equations of the stretch ratio of the deformed and undeformed radius of the spherical balloon are derived from the Lagrange equation. Firstly, the influences of the pressure in balloon and the voltage on the membrane on the equilibrium positions are discussed. The big pressure and high voltage will both induce static instability. Then, by expanding the motion equations around the static equilibrium position, the relation of the resonant frequency with the viscoelasticity are revealed. Finally, an approximation procedure is adopted, which pave the way to adopt the stochastic averaging. The probabilistic density function (PDF) of the stretch ratio around the equilibrium position is obtained. The influences of system parameters, i.e., viscoelasticity, voltage, pressure, on the probability distribution are investigated in detail. The accuracy of the proposed procedure are also been evaluated by the Monte-Carlo simulation (MCS).     

Three-dimensional elasticity solution of layered plates with viscoelastic interlayers

An analytical solution for simply supported layered plates with viscoelastic interlayers under a transverse load is proposed. The deformation of each plate layer is described by the exact three-dimensional elasticity equations. The viscoelastic property of interlayer is simulated by the generalized Maxwell model. The constitutive relation of the interlayer is simplified by the quasi-elastic approximation, which significantly simplifies the analytical process. The solution of stress and displacement fields with undetermined coefficients is derived by solving a group of ordinary differential equations. The undetermined coefficients can be efficiently deduced by using the recursive matrix technique for the plate with any number of layers. The practical convergence is observed during numerical tests. The comparison analysis indicates that the present solution has a close agreement with the finite element solution. However, the solution based on the Mindlin–Reissner hypothesis is significantly different from the present solution for thick plates. Finally, the effect of interlayer thickness on stress and displacement distributions of a five-layer plate is discussed in detail.     

A simple Kelvin and Boltzmann viscoelastic analysis of three-dimensional solids by the boundary element method

This paper presents a three-dimensional Boundary Element formulation for the analysis of simplified viscoelastic bodies without using internal cells. Two different constitutive models are considered. The first and simplest one is the Kelvin model, which does not consider instantaneous responses. The second, Boltzmann model, considers both instantaneous and viscous behaviour of materials. An appropriate kinematical relation together with differential viscoelastic constitutive representations are employed in order to built the proposed Scheme. Spatial approximations are applied for boundary elements before time solution. The proposed technique results in a time marching process that does not use relaxation (or creep) functions to recover viscous behaviour. Some examples are shown in order to demonstrate the accuracy and stability of the technique when compared to analytical solutions.     

FEM Implementation of a Three-Dimensional Viscoelastic Constitutive Model for Particulate Composites with Damage Growth

In order to analyze viscoelastic behavior of particulatecomposites with growing damage, an existing three dimensionalviscoelastic continuum damage model developed originally for solidpropellant is generalized for wider use in a Finite Element model(FEM). This equation allows for damage induced anisotropy (localtransverse isotropy). The constitutive equation is modified hereto account for the change of the material continuously froma compressible, undamaged isotropic state into the damagedanisotropic state. A fully viscoelastic time-dependentimplementation of the constitutive equation in a FEM is achievedthat allows for future extension of the FEM to simultaneously takeviscoplasticity into account. The computational results arecompared to experimental results for uniaxial and multiaxialstress states in displacement-driven experiments for solidpropellant. The multiaxial stress experiments used wide stripswith a center hole. The model predicts the experimental load andlocal strain response up to, and slightly beyond, the peak load,very well. The algorithm is shown by example to be stable far pastthe peak load.     

多自由度一般积分型粘弹性阻尼减震结构的 随机响应与等效阻尼

该文对任意多自由度带支撑一般积分型粘滞和粘弹性阻尼器减震结构的随机响应与等效阻尼比的解析分析法进行了系统研究。首先建立了结构一般运动方程;然后将运动方程化为振型广义坐标的微分和积分混合地震响应方程组;继而基于多自由度随机平均法理论,获得了结构随机平均Itô方程组的解析式,推导出耗能结构各振型振子的振幅与相位瞬态联合概率密度函数、位移与速度瞬态联合概率密度函数、位移与速度瞬态响应方差的一般解析解;最后,基于与多自由度随机平均法分析完全相同的等效准则,建立了耗能结构各振型等效阻尼比的一般解析式,并根据CQC和SRSS组合方法,建立了耗能结构随机地震响应方差的解析式,给出了带支撑广义Maxwell阻尼器和广义微分模型阻尼器减震结构随机响应和各振型等效阻尼比的一般解析式,通过与一些典型问题的模态应变能法的计算精度对比分析,表明了所提方法的有效性,使耗能结构可直接应用反应谱法进行设计,从而建立了带支撑任意线性粘滞和粘弹性阻尼器一般耗能结构随机响应与特性等效阻尼分析的完备解析解法。     

VPF粘性介质粘弹性本构关系研究

为研究粘性介质的力学性能对板材变形的影响,通过剪切蠕变-回复和松弛试验分析了甲基乙烯基粘性介质的流变性能.实验结果表明粘性介质可以简化为线性粘弹性材料.建立了粘性介质的积分型粘弹性本构方程,并结合剪切蠕变-回复过程的有限元分析确定了方程中的材料参数.利用该本构方程对粘性介质压力胀形过程进行了有限元分析,模拟结果与试验结果对比表明,所建立的本构方程可以较好的预测板材的变形过程.     

Higher-order fractional constitutive equations of viscoelastic materials involving three different parameters and their relaxation and creep functions

Two higher-order fractional viscoelastic material models consisting of the fractional Voigt model (FVM) and the fractional Maxwell model (FMM) are considered. Their higher-order fractional constitutive equations are derived due to the models’ constructions. We call them the higher-order fractional constitutive equations because they contain three different fractional parameters and the maximum order of equations is more than one. The relaxation and creep functions of the higher-order fractional constitutive equations are obtained by Laplace transform method. As particular cases, the analytical solutions of standard (integer-order) quadratic constitutive equations are contained. The generalized Mittag–Leffler function and H-Fox function play an important role in the solutions of the higher-order fractional constitutive equations. Finally, experimental data of human cranial bone are used to fit with the models given by this paper. The fitting plots show that the models given in the paper are efficient in describing the property of viscoelastic materials.     

球形沼气池模具结构振动响应的黏弹性解

考虑土和模具结构的动力相互作用, 在频率域内研究了深埋球形沼气池模具结构的动力特性。根据黏弹性理论, 通过引入势函数得到了黏弹性土体的位移、应力等的解析解。将模具结构等效为具有分数阶导数本构关系的黏弹性体, 根据黏弹性理论推导了模具结构振动响应的完全解析解。根据界面连续性条件, 确定了待定系数的具体表达式。将不同类型的黏弹性模具结构动力响应进行了对比, 并考察了分数导数本构参数、土体和模具结构各参数对系统动力特性的影响。结果表明:模具结构的黏性对系统动力响应有显著影响。随着壁厚的增加, 模具结构振动响应幅值减小;随着阶数的增加, 系统共振效应逐渐减弱;随着材料参数比的增加, 系统响应幅值减小。     

A variational formulation of constitutive models and updates in non‐linear finite viscoelasticity

The purpose of this article is to present a general framework for constitutive viscoelastic models in finite strain regime. The approach is qualified as variational since the constitutive updates obey a minimum principle within each load increment. The set of internal variables is strain‐based and employs, according to the specific model chosen, a multiplicative decomposition of strain into elastic and viscous components. The present approach shares the same technical procedures used for analogous models of plasticity or viscoplasticity, such as the solution of a minimization problem to identify inelastic updates and the use of exponential mapping for time integration. However, instead of using the classical decomposition of inelastic strains into amplitude and direction, we take advantage of a spectral decomposition that provides additional facilities to accommodate, into simple analytical expressions, a wide set of specific models. Moreover, appropriate choices of the constitutive potentials allow the reproduction of other formulations in the literature. The final part of the paper presents a set of numerical examples in order to explore the characteristics of the formulation as well as its applicability to usual large‐scale FEM analyses. Copyright © 2005 John Wiley & Sons, Ltd.     

Influence of humidity on creep response of sandwich beam with a viscoelastic soft core

Polymer foam cored sandwich beams are widely used in load-bearing components due to their high strength to weight ratio. To improve the reliability in using sandwich beams, it is essential to understand their long-term creep response in terms of variation of stresses and deformations with time under external mechanical and environmental stimuli. This paper presents an analytical model for investigating the creep response of sandwich beams made with a viscoelastic soft core, including the effect of the variable ambient humidity under the sustained load and its influence on the creep behavior. The model is based on a high-order viscoelastic structural modeling. The soft core is modeled as a viscoelastic material using differential-type constitutive relations that are based on the linear Boltzman’s principle of superposition and accounting for the deformability of the core in shear and through its thickness. Several numerical examples are presented in order to show the capability of the model and to investigate the effect of moisture on the creep behavior of sandwich beams. Finite element simulations of the creep response of sandwich beams are also performed using ABAQUS software to validate the proposed theoretical model. The results show the concentrations of shear and transverse normal stresses near the edges and their variation in time and with the change of humidity.     

基于非线性模型的沥青路面动力响应研究

根据路基变形的非线性及沥青路面具有明显粘弹性的特点,将沥青路面简化为非线性粘弹性地基上的粘弹性无限长梁,建立了移动载荷作用下非线性粘弹性梁系统动力响应数学模型。利用Adomian分解法和小波变换法得到求解稳态响应解析解的新方法。通过实际道路参数对沥青路面动力响应进行了数值仿真,研究了车速、车辆轴载、路面材料及温度对沥青路面动力响应的影响规律。结果表明该文提出的计算方法简便、快捷,是求解该类非线性动力响应问题的一种有效方法;非线性模型更能准确地反映重载及超载时沥青路面实际结构受力状态。     

黏弹性土中衬砌隧道振动响应的解析解

该文采用解析方法在频率域内对黏弹土层和衬砌结构简谐振动特性进行了研究。首先, 将土骨架视为具有分数阶导数本构关系的黏弹性体, 根据黏弹性理论, 推导得到了简谐荷载作用下分数导数型黏弹性土层的位移和应力等解析表达式。其次, 建立了两种类型的衬砌运动方程:第一, 将衬砌结构视为均匀弹性介质, 研究了分数导数黏弹性土中弹性衬砌结构的动力特性;第二, 将衬砌等效为薄壁壳体结构, 利用Flügge薄壳理论, 得到了衬砌结构的运动方程, 并对分数导数黏弹性土和壳体衬砌的动力相互作用进行了分析。根据连续性边界条件, 得到了相关待定系数的表达式。再次, 与以往的解析解进行了对比。最后, 通过算例分析了土体和衬砌各参数对系统动力特性的影响, 结果表明:薄壁壳体衬砌结构条件下系统的动力响应大于均匀弹性衬砌结构条件下系统的动力响应;随着土体和衬砌模量比的增加, 响应幅值逐渐减小。分数导数本构参数对系统的动力特性有较大影响。     

A dynamics model for nonlinear electrostrictive actuators

This paper examines the nonlinear vibration of an electrostrictive ceramic rod actuator excited by a harmonic voltage source. A frequency-domain model was developed using the nonlinear constitutive law for electrostriction. The results predict harmonic distortion of the device's displacement due to the ceramic's nonlinear behavior. AC voltage signal and DC voltage bias were studied to determine the optimum power source parameters for minimizing distortion. The calculations show that the rod's resonance frequency and amplitude depend on the electromechanical coupling strength and differ greatly for large AC voltages from the equivalent linear piezoelectric results. The nonlinear analysis relates the device's electromechanical coupling coefficient to the computed resonance and antiresonance frequencies. This important result could provide the basis for future measurement of the electrostrictive coupling coefficient using resonance techniques.     

Modeling the response of polymer–ionic liquid electromechanical actuators

A model is derived for the electromechanical response of a porous membrane swollen with an ionic liquid and sandwiched between two nanoscale-thin electrodes under DC current. Bending of the membrane is induced by pressure in pores arising due to diffusion of ions through a network of nanochannels. Transport of ions is governed by the applied electric field and redox reactions at the surfaces of electrodes. Constitutive equations for the mechanical response of a porous medium and diffusion of ions are derived by means of the free energy imbalance inequality under an arbitrary deformation with finite strains. Under the assumption regarding small strains, but finite changes in concentrations of ions and the electrostatic potential, an explicit expression is developed for the curvature of the membrane. A steady-state solution to the Poisson–Nernst–Planck equations is obtained by means of the method of matched asymptotic expansions. Results of numerical analysis demonstrate the ability of the constitutive equations to describe observations. In particular, the model provides an explanation for bending to the anode and to the cathode and predicts qualitatively the effects of applied voltage, concentration of ionic liquid, and thickness of a membrane on its curvature.     

Analytical solutions to a network of standard linear solids

Various viscoelastic models, such as the standard linear solid, Maxwell model, and Kelvin–Voigt model, are frequently used to describe the behavior of biological materials from single cells to tissues. These models are expressed mathematically as simple differential equations, called constitutive equations, which relate the applied force (stress) to the resulting deformation (strain) of the material. Networks of these models, representing materials with heterogeneous mechanical properties, are described by systems of constitutive equations. We prove that the eigenvalues associated with such systems are all nonpositive real numbers, find bounds for them, and indicate how they can be estimated quickly and accurately. We then give formulas for the analytical solutions of the system of equations.     

服从分数代数Maxwell本构模型的粘弹性阻尼材料性能分析

运用分数代数及本构理论,讨论了具有分数代数Maxwell本构关系的粘弹性材料的储能柔量、耗散柔量、耗散率、内摩擦角等参量随频率的变化规律,分析了具有分数导数Maxwell本构关系的粘弹性材料的一些阻尼特性.     

移动分布荷载下层状粘弹性体系的动力响应分析

以粘弹性Burgers模型模拟沥青面层,建立了路面结构层状粘弹性体系模型。利用积分变换和传递矩阵法,并结合广义Duhamel积分,推导出了移动分布荷载作用下层状粘弹性体系动力响应的解析解。利用样条插值函数开发了计算奇异、振荡函数多重无穷积分的计算程序,完成了动力响应从波数-频率域到时间-空间域的转化,与已有文献数据和有限元结果进行了比较,显示出了较好的一致性。结合算例分析了移动分布荷载作用下层状粘弹性体系的振动特性。结果表明,粘弹性解大于弹性解,Burgers模型能反映沥青路面变形的滞后现象,层状粘弹性体系的垂向位移随行车速度和Burgers模型材料参数的增加而减小。     

Application of Time-Based Fractional Calculus Methods to Viscoelastic Creep and Stress Relaxation of Materials

The quasi-static viscoelastic response of polymeric materialsis investigated utilizing constitutive models based on fractionalcalculus. Time-based fractional calculus analysis techniques areemphasized. Analytic solutions to quasi-static boundary value problemsin which the viscoelastic behavior is characterized by thefour-parameter fractional calculus-based solid model are given. Varioussets of data from the literature are fit with existing and newfractional calculus-based constitutive equations.     

Creep and recovery of nonlinear viscoelastic materials of the differential type

In this work, the creep and recovery properties of rubberlike viscoelastic materials in simple shear are studied by two special constitutive equations for isotropic, nonlinear incompressible viscoelastic material of the differential type. The creep and recovery processes are of significant importance to both the mechanics analysis and engineering applications. The constitutive equations introduced in this work generalize the Voigt-Kelvin solid and the 3-parameter model of classical linear viscoelasticity. They describe the uncoupled non-Newtonian viscous and nonlinear elastic response of an isotropic, incompressible material. The creep and recovery processes are treated for simple shear deformation superimposed on a longitudinal static stretch. Closed form solutions are provided and both processes are described effectively by the exponential function.