关于黏弹性材料的广义Maxwell模型

采用流变力学分析黏弹性材料的流变特性时，常要用到广义Maxwell模型 表达的应力松弛模量. 而从试验中获得的应力松弛模量，其表达式常为 Kohlrausch-William-Watts function(KWW函数)形式. 通过把KWW函数和广义Maxwell模型的拟合问题转化为两 矩阵相等的求解问题后，又把两矩阵的相等等价于两矩阵差值向量的一阶范数为无穷小的问 题，并通过引入广义逆矩阵，求得两矩阵差值向量的一阶范数的最小值，最后以一阶范数的 最小值为目标函数，松弛时间为约束条件，利用单纯形法对两矩阵差值向量的一阶范数的最 小值优化，从而提出了一种针对黏弹材料的KWW函数与广义Maxwell模型转换的计算方法. 借助于MATLAB软件，实现了对黏弹材料的广义Maxwell模型的拟合.     

基于非局部理论的分数阶黏弹性场地地震放大效应分析

基于非局部理论和分数阶导数理论,研究上覆黏弹性场地土的地震放大效应。利用Eringen非局部理论考虑土体颗粒尺度等非局部效应的影响,通过分数阶黏弹性本构模型刻画场地土的应力应变本构关系,建立基于非局部理论的分数阶黏弹性场地土的振动微分方程;考虑分数阶导数的性质和黏弹性场地土的边界条件,得到了简谐地震波作用下黏弹性场地土的位移和剪切应力的解析解,并在频率域内给出了位移放大系数和应力放大系数的表达式;最后通过数值算例分析了非局部效应、分数阶导数的阶数和土体黏性参数等对黏弹性场地地震放大效应的影响。数值分析结果表明,在低频时位移放大系数和应力放大系数随频率变化曲线存在波动,高频时逐渐趋于稳定;非局部效应对场地土位移放大系数的影响与频率有关,对应力放大系数的影响较大,在研究场地土振动效应时有必要考虑土体非局部效应的影响;分数阶导数的阶数越小,位移放大系数和应力放大系数随频率变化曲线波动越大;场地土的力学性质对场地土的振动效应的影响较大;上覆场地土的黏性对位移放大系数的影响与频率有关,高频时,土体黏性越大,位移放大系数越大;越接近基岩,土体的应力放大系数越大,且土体深度对应力放大系数的影响越大。     

非牛顿流体的广义Maxwell模型及其解

引进带分数阶导数的广义Ｍａｘｗｅｌｌ模型和Ｖｏｉｇｔ模型；用Ｌａｐｌａｃｅ变换正、反演算法，给出了非牛顿流体应力松弛和蠕变近似解析解。分数阶表征了其衰减或增长的变化特性，这是一种研究分数阶导数流变学的分析、计算方法．     

分数阶模型模拟混凝土徐变特性的多参数识别

本研究主要研究混凝土徐变特性的分数阶模型拟合及其多参数识别问题。在模型方面，提出利用修正的分数阶Maxwell模型和分数阶Poynting-Thomson模型两种模型模拟混凝土的徐变实验数据，并将结果进行对比，通过数据拟合和误差分析验证两种模型的有效性。在多参数识别方面，分别采用贝叶斯算法和布谷鸟搜索算法两种算法识别两个模型中的多个参数。研究表明：修正的分数阶Maxwell模型和分数阶Poynting-Thomson模型在刻画混凝土徐变特性中均是有效的；贝叶斯算法和布谷鸟搜索算法在分数阶模型的多参数估计问题中均是可行的，但布谷鸟算法搜索速度更快、误差更小、效率更高，在分数阶模型的多参数识别问题中性能更优。     

深埋圆柱形隧洞黏弹性衬砌-土系统稳态振动的解析解

衬砌和土体具有黏弹性性质.将土骨架和衬砌结构视为具有分数阶导数本构的黏弹性体,在频率域内研究了深埋圆柱形隧洞衬砌和土体系统的动力特性.基于黏弹性理论,根据界面连续性条件,分别得到了黏弹性土体和衬砌结构的径向位移、应力等的解析表达式.在此基础上,对比分析了经典弹性土和弹性衬砌系统、分数导数黏弹性衬砌和土体系统的动力特性.考察了土体和衬砌的模量比、衬砌厚度、分数导数阶数、材料参数比对系统动力响应的影响.结果表明:经典弹性土和弹性衬砌系统与分数导数黏弹性衬砌和土体系统的动力特性存在较大差异.随着分数导数阶数的增加,衬砌的径向位移和环向应力幅值明显减小;土体的黏性对系统动力特性的影响大于衬砌黏性的影响.     

半封闭分数导数粘弹性衬砌隧道的频域响应

混凝土衬砌具有粘弹性性质,以往的经典Kelvin模型、弹性理论和壳体理论都不能刻画其蠕变的全过程。本文基于饱和多孔介质理论,在频率域研究了轴对称荷载和流体压力作用下饱和粘弹性土中半封闭分数导数型衬砌隧洞的稳态动力响应。在引入隧洞部分透水边界条件的基础上,通过分数阶导数粘弹性模型描述衬砌的应力—位移本构关系,并利用衬砌内边界以及接触面的连续性条件,得到了饱和土和衬砌的应力、位移和孔压解答。考察了分数导数阶数、材料参数以及衬砌和土体相对渗透系数的影响。研究表明：分数导数阶数对系统响应影响较大,且依赖于衬砌的材料参数。另外,相对渗透系数对系统响应的影响很大。     

广义Maxwell黏弹性流体在两平板间的非定常流动

将分数阶微积分运算引入Maxwell黏弹性流体的本构方程，研究了黏弹性流体在两平板问的非定常流动．对于广义Maxwell黏弹性流体的分数阶导数模型，导出了对时间具有分数阶导数的特殊运动方程，利用分数阶微积分的Laplace变换理论，得到了流动的解析解．     

基于分数阶模型的牡蛎壳动力学特性研究

贝壳、牡蛎等天然材料因其轻质高强的力学特性在材料设计等领域受到了广泛的关注,但由于材料本身结构的复杂性,对其力学行为的研究十分困难。近年来,分数阶模型在研究材料的力学特性上取得了成功,相比传统模型,分数阶模型可以更好地表征复杂介质的应力或应变与时间的关系。因此,本文从波传播理论出发,以分数阶模型作为材料本构,得到了复杂介质的波传播控制方程。通过Laplace变换得到了控制方程的解析解,并通过Laplace数值逆变换分析了波的衰减对分数阶模型中参量的敏感性,讨论了不同于材料弹性、黏性的材料“惯性”特性。接着,基于解析解和多种实验测试信号,给出了得到分数阶模型参数的拟合式子。以牡蛎材料作为研究对象,利用CO₂脉冲激光器进行小试样的冲击加载、应用两点激光干涉测速系统(laser interferometer velocimetry system, VISAR)对表面粒子的速度进行测量,得到了4种密度下不同厚度的牡蛎壳试样的粒子速度时程曲线,再结合上述理论方法分析得到了牡蛎壳试样的Abel模型和分数阶Maxwell模型的参数,模型参数反映了牡蛎壳试样的细微观结构特征。结果...     

短脉冲激光加热分数阶导热及其热应力研究

短脉冲激光加热引起材料内部复杂的传热过程及热变形,现有的以Fourier定律或Cattaneo-Vernotte松弛方程结合弹性理论为框架建立起来热应力理论在刻画其热物理过程存在严重缺陷. 本文基于分数阶微积分理论, 以半空间为研究对象, 建立了分数阶Cattaneo热传导方程和相应的热应力方程, 给出了问题的初始条件和边界条件, 采用拉普拉斯变换方法, 给出了非高斯时间分布激光热源辐射下温度场和热应力场的解析解, 研究了短脉冲激光加热的温度场及热应力场的热物理行为. 数值计算中, 首先对理论解进行数值验证, 然后取分数阶变量$p=0.5$研究温度场和热应力场的变化特点及激光参数对温度和热应力的影响,最后数值计算分数阶参数对温度和热应力场的影响. 计算结果表明, 分数阶Cattaneo传热方程和热应力方程描述的温度和热应力任然具有波动特性,与经典的Fourier传热模型和标准的Cattaneo传热模型相比, 分数阶阶次越大, 热波波速越小, 热波波动性越明显; 反之, 则热波波速越大, 热扩散性越强.激光加热和冷却的速度越快, 温度上升和下降的速度越快, 压应力和拉应力交替变化越快, 温度变化幅值越小, 热应力幅值影响不明显.      

分数阶微分流变模型圆形硐室黏弹塑性分析

基于分数阶微分黏弹塑性流变模型,同时考虑受硐室掌子面影响的围岩应力释放效应,根据Laplace变换、分数阶微积分理论、Mittag-Leffler函数,推导了圆形硐室黏塑性区半径、应力、位移的理论解。将理论解与西原本构模型解进行对比,证明了分数阶模型解的合理性。分析结果表明:黏塑性区半径、洞壁环向应力的分数阶微分模型解和西原模型解随时间最终都趋于稳定,两种解法在稳定前有一定差异,稳定后大小相同;一定范围内,掌子面对围岩变形具有一定抑制作用,越靠近掌子面,黏塑性区半径和洞壁径向位移越小;流变特征对硐室围岩黏塑性区环向应力的影响较大,离硐室圆心越近,影响效果越明显,而黏塑性区径向应力和黏弹性区环向应力、径向应力受围岩流变的影响较小。     

Constitutive equation with fractional derivatives for the generalized UCM model

A fundamental problem on the constitutive equation with fractional derivatives for the generalized upper convected Maxwell model (UCM) is studied. The existing investigations on the constitutive equation are reviewed and their limitations or deficiencies are highlighted. By utilizing the convected coordinates approach, a mathematically rigorous constitutive equation with fractional derivatives for the generalized UCM model is proposed, which has an explicit expression for the stress tensor. This model can be reduced to the linear generalized Maxwell model with fractional derivatives, the UCM model and some other existing models. In addition, the rheological properties of this proposed model in the start-up of simple shear and elongation flows are investigated. It is shown that this generalized UCM model can describe the various stress evolution processes and the strain hardening effect of the viscoelastic fluids.     

Relaxation and retardation functions of the Maxwell model with fractional derivatives

A four-parameter Maxwell model is formulated with fractional derivatives of different orders of the stress and strain using the Riemann-Liouville definition. This model is used to determine the relaxation and retardation functions. The relaxation function was found in the time domain with the help of a power law series; a direct solution was used in the Laplace domain. The solution can be presented as a product of a power law term and the Mittag-Leffler function. The retardation function is determined via Laplace transformation and is solely a power law type.The investigation of the relaxation function shows that it is strongly monotonic. This explains why the model with fractional derivatives is consistent with thermodynamic principles.This type of rheological constitutive equation shows fluid behavior only in the case of a fractional derivative of the stress and a first order derivative of the strain. In all other cases the viscosity does not reach a stationary value.In a comparison with other relaxation functions like the exponential function or the Kohlrausch-Williams-Watts function, the investigated model has no terminal relaxation time. The time parameter of the fractional Maxwell model is determined by the intersection point of the short- and long-rime asymptotes of the relaxation function.     

A class of linear viscoelastic models based on Bessel functions

In this paper we investigate a general class of linear viscoelastic models whose creep and relaxation memory functions are expressed in Laplace domain by suitable ratios of modified Bessel functions of contiguous order. In time domain these functions are shown to be expressed by Dirichlet series (that is infinite Prony series). It follows that the corresponding creep compliance and relaxation modulus turn out to be characterized by infinite discrete spectra of retardation and relaxation time respectively. As a matter of fact, we get a class of viscoelastic models depending on a real parameter \(\nu > -1\). Such models exhibit rheological properties akin to those of a fractional Maxwell model (of order 1/2) for short times and of a standard Maxwell model for long times.     

Effect of the rate dependence of nonlinear kinematic hardening rule on relaxation behavior

Relaxation experiments for metallic materials and solid polymers have exhibited nonlinear dependence of stress relaxation on prior loading rate; the relaxed stress associated with the fastest prior strain rate has the smallest magnitude at the end of the same relaxation periods. Modeling capability for the basic feature of relaxation behavior is qualitatively investigated in the context of unified state variable theory. Unified constitutive models are categorized into three general classes according to the rate dependence of kinematic hardening rule, which defines the evolution of the back (equilibrium) stress and is the major difference among constitutive models. The first class of models adopts the nonlinear kinematic hardening rule proposed by Armstrong and Frederick. In this class, the back stress appears to be rate-independent under loading and subsequent relaxation conditions. In the second class of models, a stress rate term is incorporated into the Armstrong–Frederick rule and the back stress then becomes rate-dependent during relaxation condition even though it remains rate-independent under loading condition. The final class proposed here includes a new nonlinear kinematic hardening rule that causes the back stress to be rate-dependent all the time. It is shown that the apparent rate dependence of the back stress during relaxation enables constitutive models to predict the influence of prior loading rate on relaxation behavior.     

Rheological representation of fractional order viscoelastic material models

We develop rheological representations, i.e., discrete spectrum models, for the fractional derivative viscoelastic element (fractional dashpot or springpot). Our representations are generalized Maxwell models or series of Kelvin-Voigt units, which, however, maintain the number of parameters of the corresponding fractional order model. Accordingly, the number of parameters of the rheological representation is independent of the number of rheological units. We prove that the representations converge to the corresponding fractional model in the limit as the number of units tends to infinity. The representations extend to compound fractional derivative models such as the fractional Maxwell model, fractional Kelvin-Voigt model, and fractional standard linear solid. Computational experiments show that the rheological representations are accurate approximations of the fractional order models even for a small number of units.     

Non-linear viscoelastic behavior of polymer melts interpreted by fractional viscoelastic model

Very recently, researchers dealing with constitutive law pertinent viscoelastic materials put forward the successful idea to introduce viscoelastic laws embedded with fractional calculus, relating the stress function to a real order derivative of the strain function. The latter consideration leads to represent both, relaxation and creep functions, through a power law function. In literature there are many papers in which the best fitting of the peculiar viscoelastic functions using a fractional model is performed. However there are not present studies about best fitting of relaxation function and/or creep function of materials that exhibit a non-linear viscoelastic behavior, as polymer melts, using a fractional model. In this paper the authors propose an advanced model for capturing the non-linear trend of the shear viscosity of polymer melts as function of the shear rate. Results obtained with the fractional model are compared with those obtained using a classical model which involves classical Maxwell elements. The comparison between experimental data and the theoretical model shows a good agreement, emphasizing that fractional model is proper for studying viscoelasticity, even if the material exhibits a non-linear behavior.     

基于Riesz势空间分数阶算子的非局部粘弹性力学元件

将幂函数引入Eringen非局部线粘弹性本构,导出Riesz势形式的应力-应变关系。利用该关系,构造非局部弹簧和非局部阻尼器两类元件;利用元件的串联和并联,建立非局部Kelvin和非局部Maxwell粘弹性模型,推导模型的松弛模量和蠕变柔量。进一步,给出非局部粘弹性模型在生物组织超声波耗散建模中的应用。     

A Numerical Study of Oscillating Peristaltic Flow of Generalized Maxwell Viscoelastic Fluids Through a Porous Medium

This article presents a numerical study on oscillating peristaltic flow of generalized Maxwell fluids through a porous medium. A sinusoidal model is employed for the oscillating flow regime. A modified Darcy-Brinkman model is utilized to simulate the flow of a generalized Maxwell fluid in a homogenous, isotropic porous medium. The governing equations are simplified by assuming long wavelength and low Reynolds number approximations. The numerical and approximate analytical solutions of the problem are obtained by a semi-numerical technique, namely the homotopy perturbation method. The influence of the dominating physical parameters such as fractional Maxwell parameter, relaxation time, amplitude ratio, and permeability parameter on the flow characteristics are depicted graphically. The size of the trapped bolus is slightly enhanced by increasing the magnitude of permeability parameter whereas it is decreased with increasing amplitude ratio. Furthermore, it is shown that in the entire pumping region and the free pumping region, both volumetric flow rate and pressure decrease with increasing relaxation time, whereas in the co-pumping region, the volumetric flow rate is elevated with rising magnitude of relaxation time.     

Shear relaxation characteristics of rock joints under stepwise loadings

The stress relaxation characteristic of rock mass is an important aspect of rheology and has important practical significance for rock engineering. In order to investigate the relaxation characteristic of rock joints with different slope ratios and normal stresses, a series of shear stress relaxation tests were conducted on artifical rock joints poured by cement mortar. Test results show that the relaxation curves can be divided into three stages, i.e. instantaneous relaxation stage, attenuation relaxation stage, and stable relaxation stage. Furthermore, the nonlinear Maxwell relaxation equation was obtained by using the relation between the viscosity coefficient and time, and the theoretical curves based on the empirical equation agreed well with the test results. Moreover, the change law of the initial viscosity coefficient was investigated. Accordingly, a stress relaxation method, termed as relaxation stress peak method, was proposed to determine the long-term strength of rock joints.