In view of the increasing attention to the time responses of complex fluids described by power-laws in association with the need to capture inertia effects that manifest in high-frequency microrheology, we compute the five basic time-response functions of in-series or in-parallel connections of two elementary fractional derivative elements known as the Scott-Blair (springpot) element. The order of fractional differentiation in each Scott-Blair element is allowed to exceed unity reaching values up to 2 and at this limit-case the Scott-Blair element becomes an inerter—a mechanical analogue of the electric capacitor that its output force is proportional only to the relative acceleration of its end-nodes. With this generalization, inertia effects may be captured beyond the traditional viscoelastic behavior. In addition to the relaxation moduli and the creep compliances, we compute closed-form expressions of the memory functions, impulse fluidities (impulse response functions) and impulse strain-rate response functions of the generalized fractional derivative Maxwell fluid, the generalized fractional derivative Kelvin-Voigt element and their special cases that have been implemented in the literature. Central to these calculations is the fractional derivative of the Dirac delta function which makes possible the extraction of singularities embedded in the fractional derivatives of the two-parameter Mittag-Leffler function that emerges invariably in the time-response functions of fractional derivative rheological models.

中文翻译：

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分数阶导数流变模型的时间响应函数

鉴于人们越来越关注幂律描述的复杂流体的时间响应以及需要捕获高频微流变学中表现出的惯性效应，我们计算了串联或串联的五个基本时间响应函数- 两个基本分数阶导数元素的平行连接，称为 Scott-Blair (springpot) 元素。每个 Scott-Blair 元件中的分数阶微分阶数允许超过统一值，达到 2 并且在这种极限情况下，Scott-Blair 元件变成了惰性——电容器的机械模拟，其输出力仅成比例其末端节点的相对加速度。通过这种概括，惯性效应可能会超出传统的粘弹性行为。除了松弛模量和蠕变柔量之外，我们还计算了广义分数阶导数麦克斯韦流体、广义分数阶导数 Kelvin- 的记忆函数、脉冲流动性（脉冲响应函数）和脉冲应变率响应函数的闭式表达式。 Voigt 元素及其在文献中已实现的特殊情况。这些计算的核心是 Dirac delta 函数的分数阶导数，这使得提取嵌入在二参数 Mittag-Leffler 函数的分数阶导数中的奇点成为可能，该函数总是出现在分数阶导数流变模型的时间响应函数中。广义分数阶导数麦克斯韦流体的脉冲流动性（脉冲响应函数）和脉冲应变率响应函数，广义分数阶导数 Kelvin-Voigt 单元及其在文献中实现的特殊情况。这些计算的核心是 Dirac delta 函数的分数阶导数，这使得提取嵌入在二参数 Mittag-Leffler 函数的分数阶导数中的奇点成为可能，该函数总是出现在分数阶导数流变模型的时间响应函数中。广义分数阶导数麦克斯韦流体的脉冲流动性（脉冲响应函数）和脉冲应变率响应函数，广义分数阶导数 Kelvin-Voigt 单元及其在文献中实现的特殊情况。这些计算的核心是 Dirac delta 函数的分数阶导数，这使得提取嵌入在二参数 Mittag-Leffler 函数的分数阶导数中的奇点成为可能，该函数总是出现在分数阶导数流变模型的时间响应函数中。