【作者】 尹义林；

【导师】 杨正宏；

【作者基本信息】 同济大学 ， 材料科学与工程， 2022， 博士

【摘要】 沥青及沥青混合料是典型的黏弹性材料,其力学行为具有显著的温度和时间(或频率)依赖性,通常用经验数学模型和力学模型描述其线性黏弹性,大部分模型根据频域方法提出。现在公认的能最好地描述其动态力学行为的力学模型是2S2P1D模型,它由两个弹簧、两个抛物线型蠕变元件(分数阶导数元件)和一个黏壶组成,其动态响应函数很复杂,难以通过积分变换推导出模型的瞬变函数。有些与试验数据拟合较好的经验数学模型仅含有复模量模函数,而无相角函数,无法计算其储能模量和损耗模量函数,它们对黏弹性力学行为的描述不完整。因此,有必要采用简化模型或解析表达式描述沥青混合料的瞬变和动态力学行为,并提出简单的模型参数求解方法。本文的主要研究内容和结果包括:首先分析了多个沥青及沥青混合料线性黏弹性模型的动态响应函数的特征,讨论了模型描述力学行为的优缺点。然后叙述了简单力学模型、广义Maxwell和Kelvin–Voigt模型以及分数阶导数模型的本构方程、瞬变和动态响应函数,讨论了这些力学模型的动态响应函数特征。此外阐述了黏弹性函数之间的精确计算关系,包括:瞬变和动态响应函数的关系,瞬变函数松驰模量和蠕变柔量的关系,动态响应函数的组成部分的Kramers–Kronig积分关系。主要试验和分析方法包括:测定沥青结合料的基本物理力学性能和玻璃化转变温度;采用动态力学分析(DMA)方法分析沥青及沥青混合料的动态力学行为,通过频率扫描方法和时温叠加原理得到动态力学试验数据;通过弯曲蠕变试验分析沥青混合料的静态力学行为;应用Kramers–Kronig变换分析仅含有复模量模的经验数学模型的动态响应函数和实测动态力学试验数据。建立力学模型的复动态函数的实部和虚部的关系。简单力学模型(Maxwell、Kelvin–Voigt、标准线性固体模型)的模量或柔量复平面图是圆心在实轴的完整半圆;分数阶导数模型(Maxwell、Kelvin–Voigt、标准线性固体模型)的模量或柔量复平面图是圆心在第四象限的偏转半圆。由于试验仪器能达到的频率范围有限,从实际动态试验的试验结果中仅能观察到一部分圆弧。力学模型元件参数可容易地通过回归半圆与实轴的交点和圆心位置(或圆半径)求解。材料的黏弹性通常用弹簧、黏壶和分数阶导数元件串并联组成的力学模型形象地表示,这仅是模拟黏弹性的理想假设方法。本文试图从系统内部的未知因素出发,更一般地讨论黏弹性的内在根源;从材料在应力作用下产生的密度梯度观点对分数阶导数元件提出物理解释。建立力学模型的复动态函数解析表达式的一般形式。根据现代控制工程的线性动态系统的传递函数方法,材料的黏弹性可用正弦传递函数表示,正弦传递函数的增益、特征时间常数和因子的幂可容易地通过复模量模的Bode图(主曲线)的渐近线求解。利用频域分析的模量复平面图和复模量模的Bode图,以及利用时域分析的弯曲蠕变柔量试验结果分别求解沥青混合料的模型参数,沥青混合料的动态力学和瞬变试验结果均表明,分数阶标准线性固体模型可较好地描述沥青混合料的动态和静态力学行为。动态力学分析的复动态函数的实部和虚部在数学上不相互独立,模和相角也不相互独立,它们之间通过Kramers–Kronig变换联系。应用Kramers–Kronig变换可得到仅含有复模量模的经验数学模型的其他动态响应函数,可补充实测动态力学试验数据,可检验动态力学试验数据的有效性。更多还原

【Abstract】 Bituminous binders and mixtures are the typical members in the group of viscoelastic materials of which the mechanical behavior strongly relates to temperature and time(or frequency).Linear viscoelasticity is usually described by empirical mathematical models or mechanical models,most of which are established from frequency domain.The 2S2P1 D model,an abbreviation of the combination of two springs,two parabolic creep elements(fractional derivative elements)and one dashpot,is so far recognized as a mechanical model which can best characterize the dynamic mechanical behavior.However,its dynamic response function is too complicated to deduce the transient function via integral transformations.Some other empirical mathematical models without phase angle function which can well describe experimental data are only formulas for the complex modulus magnitude.It seems incomplete for the description of viscoelasticity since the storage and loss functions cannot be obtained.Thus it is necessary to find a simplified model or an analytical expression to describe the transient and dynamic mechanical behavior of bituminous mixtures,and propose simple methods to determine model parameters.The main contents and results of this dissertation are as follows:We first analyze the characteristics of dynamic response functions of several linear viscoelastic models for bituminous binders and mixtures,and discuss the advantages and disadvantages of these models in describing mechanical behavior.Then,we review the constitutive equations,transient and dynamic response functions of simple mechanical models,generalized Maxwell and Kelvin–Voigt models,and fractional derivative models.The characteristics of dynamic response functions of mechanical models in the dynamic mechanical analysis(DMA)are discussed.In addition,exact interrelations among the viscoelastic functions are introduced,including the interrelation of a transient with the corresponding dynamic functions,the interrelation of the two transient functions,and interrelations between the components of a complex dynamic function,known as Kramers–Kronig integral relations.The main experiments and analysis methods are listed as follows.The classical physical properties and glass transition temperature of bituminous binders are measured.The dynamic mechanical analysis(DMA)is applied to characterize the dynamic mechanical behavior of bituminous binders and mixtures.The dynamic mechanical data are obtained with the aid of frequency sweep and the time-temperature superposition principle(TTSP).The static mechanical behavior of bituminous mixtures is measured by bending creep tests.Kramers–Kronig transforms are used to analyze dynamic response functions of an empirical mathematical model containing only complex modulus magnitude function and measured dynamic mechanical data.The relation between real and imaginary parts of complex dynamic function of a mechanical model has been established.For the simple mechanical models,including Maxwell,Kelvin–Voigt,and standard linear solid models,the complex plane plot of modulus or compliance is a full semicircle with its center on the real axis.For the fractional derivative models(Maxwell,Kelvin–Voigt,and standard linear solid models),the complex plane plot is a depressed or distorted semicircle with its center below the real axis.Only a part of the semicircle can be seen,since experimental data are collected over restricted frequency ranges due to the instrumental limitations in actual dynamic measurements.The model element parameters can be easily determined by the two intercepts of the extrapolated circular arc with the real axis and the displacement of the semicircle center(or radius of the circle)in the complex plane plot.Viscoelastic behavior of materials is usually characterized by a mechanical model composed of spring and dashpot or fractional derivative elements in series and parallel combination,which is only an ideal assumption method to simulate viscoelasticity.This dissertation attempts to discuss the nature of viscoelasticity with unknown factors in a system.A physical interpretation for the fractional derivative element is proposed from the view of the density gradient of a material produced by stress.We have established a general form of analytical expressions for complex dynamic functions of mechanical models.From a viewpoint of sinusoidal transfer function of a linear dynamic system in modern control engineering,viscoelasticity of materials can also be characterized by a sinusoidal transfer function,in which the gain,characteristic time constants and powers of the factors can be readily identified by the asymptotes of Bode plot(master curve)of complex modulus magnitude.In the frequency domain,the model parameters for bituminous mixtures are identified graphically via the complex plane plot and Bode plot of magnitude;in the time domain,the model parameters are determined by the bending creep compliance curve.Both the experimental results of dynamic and transient experiments show that the dynamic and static mechanical behavior of bituminous mixtures can be adequately described by the fractional derivative standard linear solid model.The real and imaginary parts in a complex dynamic function are not mathematically independent of each other,and its magnitude and phase angle are also not independent.They are connected by Kramers–Kronig transforms,which are expected to obtain other dynamic response functions of an empirical mathematical model containing only complex modulus magnitude function,complete measured dynamic mechanical data,and check the validity of dynamic mechanical data.更多还原