聚合物应力松弛的非定常微分型本构模型EI北大核心CSCD

利用粘滞系数随时间变化的粘性元件和弹性模量随时间变化的弹性元件,构造非定常(也可称为变参数)Maxwell模型和非定常Zener模型。求解非定常模型的本构方程得到它们的松弛函数。结果表明,当粘滞系数和弹性模量随时间按幂律规律变化时,可以把经验函数stretched exponential函数和修正的stretched exponential函数视为非定常模型的应力松弛函数。文中用修正的stretched exponential函数对聚甲基丙烯酸甲酯(PMMA)和聚四氟乙烯(PTFE)松弛模量实验数据进行了拟合,表明该函数能较好地描述这两种聚合物的应力松弛。     

聚合物应力松弛的非定常微分型本构模型

利用粘滞系数随时间变化的粘性元件和弹性模量随时间变化的弹性元件,构造非定常(也可称为变参数)Maxwell模型和非定常Zener模型。求解非定常模型的本构方程得到它们的松弛函数。结果表明,当粘滞系数和弹性模量随时间按幂律规律变化时,可以把经验函数stretched exponential函数和修正的stretched exponential函数视为非定常模型的应力松弛函数。文中用修正的stretched exponential函数对聚甲基丙烯酸甲酯(PMMA)和聚四氟乙烯(PTFE)松弛模量实验数据进行了拟合,表明该函数能较好地描述这两种聚合物的应力松弛。     

聚合物的水声声衰减能力与动态力学性能的关系

以描述高分子材料粘弹行为的三元模型为出发点,以声波在高分子介质中的传播理论为依据,推导出了材料的水声声衰减能力与材料的动态力学性能参数包括损耗因子、松弛前的剪切模量、松弛后的剪切模量以及材料的密度和厚度之间的关系式。为实验验证所推导的关系式,设计合成了一系列阻尼性能不同的聚合物,分别测试了它们的动态力学性能和声衰减能力。用材料的动态力学性能参数计算得到的水声声衰减系数与实验测得的声衰减能力相符合。该数学模型为找出吸声系数与材料的动态力学性能之间的关系、指导水声材料设计奠定了基础。     

新型BTG塑料合金关于温度-频率-振幅的动态阻尼性能模型

建立了描述一种新型BTG塑料合金的温度-频率-振幅的动态阻尼性能数学模型。通过动态热分析仪DMA242, 获取了BTG塑料合金的频率扫描、温度扫描和幅值扫描的动态阻尼损耗因子实验数据。通过分析实验数据, 将温度扫描的阻尼损耗因子量分离为仅与频率相关以及与频率、温度均相关的两个分量, 并分别用Kelvin分数导数的阻尼损耗因子模型和高斯函数模型来表达这两个分量, 再以此为基准, 考虑振幅对阻尼损耗的影响, 由此建立了综合考虑温度-频率-振幅的阻尼损耗数学模型。结果表明, 所建立的综合考虑温度-频率-振幅的阻尼损耗数学模型能准确描述实验数据。      

分数阶导数线性流变固体模型及其应用

采用Koeller 弹壶元件替代标准线性固体模型中的Newton黏壶, 得到分数阶导数线性流变固体模型, 给出了表征模型动态黏弹特性的存储模量、损耗模量和损耗因子以及表征模型静态黏弹特性的蠕变柔量和松弛模量。采用分数阶导数线性流变固体模型、标准线性固体模型和五参量固体模型对聚丙烯材料应力松弛特性进行分析。讨论了Mittag-Leffler函数的求和截断误差。结果表明分数阶导数线性流变固体模型能更准确描述聚丙烯材料的应力松弛行为。     

苯基硅橡胶的动态力学性能研究

用橡胶加工分析仪(RPA2000)表征了SE系列苯基硅橡胶在不同应变、温度、频率下的动态力学性能。结果表明:在不同应变下,SE系列硫化胶随应变的增大弹性模量G′减小,损耗因子tanδ略有上升,混炼胶的黏性扭矩S″增大;在不同温度下,损耗因子tanδ随温度的升高而降低;在不同频率下,混炼胶的损耗因子tanδ随频率的升高而降低,硫化胶的损耗因子tanδ随频率的升高而升高,混炼胶的损耗因子比硫化胶的损耗因子大得多。     

气辅挤出胀大比与滑移段长度的关系

采用参数渐变法和Thompson变换,对粘弹性高分子熔体在不同气体辅助挤出口模内的流动进行了数值模拟研究。考察了体积流量、松弛时间和滑移段长度对挤出物挤出胀大比的影响。研究表明,熔体在滑移段的停留时间与材料松弛时间之比与挤出胀大比之间存在指数衰减关系,其实质是熔体在滑移段处于形变衰减过程。理论分析与数值模拟具有高度的一致性,表明该方程可用于指导气辅口模设计。     

复合材料粘弹性本构关系与热应力松弛规律研究Ⅱ:数值分析

给出了预测复合材料粘弹性松弛模量、等效热应力松弛系数和等效时变热膨胀系数的均匀化方法的有限元数值实现步骤, 研究了单向纤维复合材料随温度变化的粘弹性本构关系, 以及热应力松弛规律和热膨胀系数的时变特征。单向纤维复合材料的一维热变形分析数据显示了热应变对时间的强烈依赖关系;以数值形式给出的等效热应力松弛模量对时间的依赖关系表明, 等效的热应力松弛模量对时间的依赖性较弱, 其冲击模量和渐近模量只相差0.4 %。     

考虑裂纹表面摩擦阻尼的振动疲劳裂纹扩展分析

以含表面裂纹悬臂梁为研究对象,研究了裂纹面摩擦效应对裂纹疲劳扩展的影响。分析时,用双线性弹簧描述裂纹呼吸行为,用Galerkin方法把呼吸裂纹梁简化为单自由度系统,基于Coulomb摩擦模型和能量耗散理论推导了摩擦阻尼损耗因子,运用广义的Forman方程模拟疲劳裂纹扩展,通过振动分析与裂纹扩展计算同步进行的方法考虑振动与疲劳的耦合效应,探讨了摩擦阻尼对裂纹梁疲劳裂纹扩展寿命的影响。结论表明,摩擦阻尼损耗因子随裂纹扩展呈单调递增趋势,摩擦阻尼对振动疲劳裂纹扩展的影响不容忽视     

基于不确定性的液压泵壳体温度变化规律研究

针对液压泵壳体缺少更加符合工程实际的温度变化定量研究问题,首先建立确定性状态下的热力学模型,然后根据随机因子法及求解函数数字特征的代数综合法,建立随机不确定性状态下的热力学模型.对两种热力学模型下的温度随时间变化规律进行了研究,得到温度期望随时间增加而变大,均方差随时间增加而变小的规律,拓展了液压泵热力学研究领域.     

Fitting stress relaxation experiments with fractional Zener model to predict high frequency moduli of polymeric acoustic foams

This paper presents a time domain method to determine viscoelastic properties of open-cell foams on a wide frequency range. This method is based on the adjustment of the stress–time relationship, obtained from relaxation tests on polymeric foams’ samples under static compression, with the four fractional derivatives Zener model. The experimental relaxation function, well described by the Mittag–Leffler function, allows for straightforward prediction of the frequency-dependence of complex modulus of polyurethane foams. To show the feasibility of this approach, complex shear moduli of the same foams were measured in the frequency range between 0.1 and 16 Hz and at different temperatures between ?20 °C and 20 °C. A curve was reconstructed on the reduced frequency range (0.1 Hz–1 MHz) using the time–temperature superposition principle. Very good agreement was obtained between experimental complex moduli values and the fractional Zener model predictions. The proposed time domain method may constitute an improved alternative to resonant and non-resonant techniques often used for dynamic characterization of polymers for the determination of viscoelastic moduli on a broad frequency range.     

Visco-elastic behavior through fractional calculus: An easier method for best fitting experimental results

In capturing visco-elastic behavior, experimental tests play a fundamental rule, since they allow to build up theoretical constitutive laws very useful for simulating their own behavior. The main challenge is representing the visco-elastic materials through simple models, in order to spread their use. However, the wide used models for capturing both relaxation and creep tests are combinations of simple models as Maxwell and/or Kelvin, that depend on several parameters for fitting both creep and relaxation tests. This paper, following Nutting and Gemant idea of fitting experimental data through a power law function, aims at stressing the validity of fractional model. In fact, as soon as relaxation test is well fitted by power law decay then the fractional constitutive law involving Caputo’s derivative directly appears. It will be shown that fractional model is proper for studying visco-elastic behavior, since it may capture both relaxation and creep tests, requiring the identification of two parameters only. This consideration is assessed by the good agreement between experimental tests on creep and relaxation and the fractional model proposed. Experimental tests, here reported are performed on two polymers having different chemical physical properties such that the fractional model may cover a wide range of visco-elastic behavior.     

Universal linear viscoelastic approximation property of fractional viscoelastic models with application to asphalt concrete

The paper presents a comprehensive linear viscoelastic characterization of asphalt concrete using fractional viscoelastic models. For this purpose, it is shown that fractional viscoelastic models are universal approximators of relaxation and retardation spectra. This essentially means that any spectrum can be mathematically represented by fractional viscoelastic models. Characterization of asphalt concrete is performed by constructing the dynamic modulus master curve and determining the parameters of the generalized fractional Maxwell model (GFMM). This procedure is similar to the widely used one of determining the master curve of asphalt concrete using a statistical function such as the sigmoidal model. However, from the GFMM, the relaxation modulus, creep compliance, continuous relaxation spectrum, and Prony series parameters can be determined analytically. A further advantage of the GFMM is that unlike the sigmoidal model, which only gives a representation of either the dynamic modulus or the storage modulus, the GFMM gives a representation of both the storage modulus and loss modulus (and therefore also the dynamic modulus and phase angle). The procedure was successfully applied to ten different mixes used in the State of Virginia.     

Combination of a standard viscoelastic model and fractional derivate calculus to the characterization of polymers

Polymeric materials are known to be more or less dispersive and absorptive. In the field of mechanical vibrations, dispersion has for consequence that the dynamic modulus is frequency dependent, and absorption is exhibited by the fact that these materials have the ability to absorb energy under vibratory motion. The phenomenon of dispersion in conjunction with the notion of complex Modulus of Elasticity, permits to establish the relation between the real and the imaginary components of the Modulus of Elasticity, i.e. respectively the dynamic and loss moduli. The loss factor is simply determined through taking the quote of these two components of the Modulus of Elasticity. The theoretical background for the interrelations between the dynamic modulus and the loss modulus is found in the Kramers–Kronig relations. However, and due to the mathematical difficulties encountered in using the exact expressions of these relations, approximations are necessary for applications in practical situations. On the other hand, several simple models have been proposed to explain the viscoelastic behaviour of materials, but all fail in giving a full account of the phenomenon. Among these models, the standard viscoelastic model, or more known as the Zener model, is perhaps the most attractive one. To improve the performance of this model, the concept of fractional derivates has been incorporated into it, and which results in a four-parameter model. Applications have also shown the superiority of this model when theoretical predictions are compared to experimental data of different polymeric materials. The aim of this paper is to present the results of applying this model to rubber, both natural and filled, and to some other selected more general polymer. Electronic Publication     

Augmented Hooke's law based on alternative stress relaxation models

 Relationships are presented which are considered to be the most general, linear, constitutive stress–strain relationships for a simple, solid material at isothermal conditions. Formally, they are obtained in frequency domain by adding “anelastic”, frequency dependent terms to the elastic constants of the material. The resulting, basic, augmented Hooke's law (AHL) is proposed as a general framework for comparison of alternative linear material damping models. Contained, as special cases, are both the classical, purely mechanical theory of viscoelasticity and more recent damping models based on linear, irreversible thermodynamics. The original AHL Helmholtz free energy density function, Dovstam (1995), is generalised to materials with continuously distributed relaxation frequency spectra. It is shown that there corresponds a continuously distributed relaxation spectrum to each admissible linear damping model and how such relaxation spectra may be computed using the Stieltjes–Perron inversion formula and explicit (analytical) models of the complex, frequency dependent, parts of a corresponding AHL. Traditional relaxation time spectra (discrete or continuously distributed relaxation times) are shown to be directly related to AHL relaxation amplitude distributions (relaxation frequency spectra) derived in the paper. The relationships between traditional relaxation time spectra and frequency domain AHL relaxation amplitude distributions connect experimental time domain data in linear viscoelasticity, with corresponding AHL relaxation frequency spectra which may be used in linear, constitutive material damping modelling. The results indicate that the information supplied by relaxation spectra (in time or frequency domain) is completely equivalent to any suitable and physically realistic damping model, properly curve fitted to experimental complex material moduli. Fractional derivative models are demonstrated to simulate the mean properties of the relaxation processes in the material during vibration. In this context, fractional derivative models are completely equivalent to frequency domain, continuously distributed AHL relaxation models, with well defined and easily computed relaxation frequency spectra. Using experimentally estimated data, it is explicitly demonstrated that linear, material damping may be simulated using discrete as well as continuously distributed AHL relaxation models or corresponding fractional derivative models. Which damping model to use is a matter of convenience. Received 17 December 1999     

硫化橡胶动态力学性能的分数阶微分Kelvin模型

研究炭黑填充硫化橡胶的动态粘弹性,采用Gabo Eplexor 500N对材料进行不同频率时的温度扫描测试,得到材料玻璃化转变温度Tg随频率的变化规律。在Tg~Tg+50℃范围内进行不同温度的频率扫描测试,得到材料存储模量、损耗模量和损耗因子。采用分数阶微分Kelvin模型对动态粘弹特性进行分析,确定了模型参数。结果表明,分数阶微分Kelvin模型可以较好地描述材料在不同温度和较宽频率范围内的动态粘弹性力学行为。当温度高于Tg时,随着温度的升高,材料从Tg附近的粘弹态向高温时的橡胶态转变,模型中的微分阶数相应地逐渐减小。     

Ultrasonic Determination of Stiffness Properties of an Orthotropic Viscoelastic Material

Abstract

Ultrasonic velocity measurements were used to populate the real portion of the stiffness matrix of an orthotropic viscoelastic material, as a function of frequency. The orthotropy originated from short cellulose reinforcing fibers cast into the polypropylene matrix. The results were consistent with a Zener relaxation model for the material, with a time constant of the order of a few microseconds. It was found that the shape of the dispersion curve varied markedly with orientation: The frequency dependence of the phase velocity was at a minimum along the primary axis of the reinforcing fibers; the fibers were shown to inhibit viscoelastic behavior in this direction. The attenuation coefficient was measured along one primary axis of the material, and found to be consistent with the dispersion curves and localized form of the Kramers-Kronig relationships linking the real and imaginary components of the stiffness matrix.     

Ultrasonic Determination of Stiffness Properties of an Orthotropic Viscoelastic Material

Ultrasonic velocity measurements were used to populate the real portion of the stiffness matrix of an orthotropic viscoelastic material, as a function of frequency. The orthotropy originated from short cellulose reinforcing fibers cast into the polypropylene matrix. The results were consistent with a Zener relaxation model for the material, with a time constant of the order of a few microseconds. It was found that the shape of the dispersion curve varied markedly with orientation: The frequency dependence of the phase velocity was at a minimum along the primary axis of the reinforcing fibers; the fibers were shown to inhibit viscoelastic behavior in this direction. The attenuation coefficient was measured along one primary axis of the material, and found to be consistent with the dispersion curves and localized form of the Kramers-Kronig relationships linking the real and imaginary components of the stiffness matrix.     

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.