沥青混合料动态黏弹性分数阶导数模型

Fractional Derivative Model for Dynamic Viscoelasticity of Asphalt Mixtures

为了更加精确、全面地表征沥青混合料的动态黏弹性力学特性,基于四参数分数阶导数Zener模型(FDZ)分配微分算子后,分别得到修改五参数分数阶导数模型(MFPFD)和改进五参数分数阶导数模型(IFPFD)的本构方程,并进一步得到了模型的复数模量解析表达式。将复数模量分离实部和虚部得到存储模量、损失模量及损耗因子的系数解析表达式。采用LARE目标函数,基于时温等效原理,构造了3个模型的黏弹函数主曲线,通过拟合优度和目标函数最优值评价模型拟合的效果,并对2种分数阶导数模型进行了对比。研究结果表明:FDZ模型和MFPFD模型黏弹函数呈现相似的性质,说明FDZ模型分配微分算子后黏弹函数的特性未发生变化;IFPFD模型采用1套参数即可较好地表征沥青混合料的所有动态黏弹函数,如存储模量、损失模量、相位角、动态模量、损耗因子及Cole-cole曲线,满足Kramers-Kronig关系;与FDZ模型相比,IFPFD模型增加模型参数β,能够更好地描述损失模量的峰宽和其他黏弹函数的非对称特征。最后,IFPFD模型的参数具有一定的物理意义,而根据数值优化所得的模型参数,能确定参考温度下动态黏弹性分数阶导数的微分方程,且分数阶微分本构方程较为简单,研究结果可为沥青混合料设计和路面动力学分析提供新的思路。

To describe the dynamic viscoelastic behavior of asphalt mixture more accurately and comprehensively,the constitutive equation of the modified five-parameter fractional derivative model(MFPFD)and improved five-parameter fractional derivative model(IFPFD)were obtained based on the four-parameter fractional derivative Zener model(FDZ)by assigning differential operators.At the same time,the analytical expression of complex modulus for fractional derivative model was obtained.the analytical expressions of storage modulus,loss modulus and loss factor were obtained by separating the real part and the imaginary part of analytical expression of complex modulus.The master curve of the model was constructed based on the time-temperature equivalence principle by using the LARE objective function.The fitting results of the model was evaluated by the goodness-of-fit and the optimal value of the objective function,and three fractional derivative models were compared.The results show that the characteristics of viscoelastic functions for FDZ model and MFPFD model have similar properties,which indicate that the characteristics of viscoelastic functions for FDZ model do not change after assigning differential operators;The IFPFD model can better describe all dynamic viscoelastic functions of asphalt mixtures by using one set of parameters,such as storage modulus,loss modulus,phase angle,dynamic modulus,loss factor and Cole-cole curve,which satisfy the Kramers-Kronig relationship.Compared with FDZ model,due to the addition of model parameterβ,the IFPFD model can better describe the peak width of loss modulus and the asymmetry characteristics of viscoelastic functions.Moreover,the coefficients of the model have certain physical significance,the differential equation of dynamic viscoelastic fractional derivative at specific temperature can be determined according to the model parameters obtained by numerical optimization.And the fractional differential constitutive equation is relatively simple,which would be provided a new insight for asphalt mixture design and pavement dynamic analysis.