3. Viscoelastic models based on the Mittag-Leffler function

## 3.1. The fractional K-BKZ model

We first note that the Maxwell-Debye relaxation of stress (exponential decay—see Eqs. (4) and (5)) is quite common, but there are many real materials showing different types of fading memory, such as <sup>a</sup> power law decay G tðÞ� <sup>t</sup> , <sup>0</sup> <sup>&</sup>lt; <sup>α</sup> <sup>&</sup>lt; <sup>1</sup> [8]. For example, the critical gel �<sup>α</sup> model investigated by Winter and Chambon is written G tðÞ¼ St . If we assume the relaxa- �<sup>α</sup> tion modulus for an arbitrary loading history in such materials is given by Gðt � t 0 Þ ¼ �<sup>1</sup> � <sup>V</sup>ðΓð<sup>1</sup> � <sup>α</sup>ÞÞ <sup>ð</sup><sup>t</sup> � <sup>t</sup> 0 <sup>Þ</sup> <sup>α</sup> (<sup>V</sup> is known as <sup>a</sup> quasi-property [9] and is connected to the critical gel strength by S ¼ V=Γð1 � αÞ), then we have that:

$$\mathfrak{w}(t) = \frac{1}{\Gamma(1-\alpha)} \int\_0^t \mathbb{V}(t-t')^{-\alpha} \frac{d\chi}{dt'} dt'.\tag{11}$$

By recognizing that the Caputo fractional derivative of a general function γð Þt (in our case γð Þt is the deformation) is defined as [10]:

$$\frac{d^\alpha \mathbf{y}(t)}{dt^\alpha} = \frac{1}{\Gamma(1-\alpha)} \int\_0^t (t-t')^{-\alpha} \frac{d\mathbf{y}}{dt'} dt',\tag{12}$$

we obtain a generalized viscoelastic model [10, 11], that can be written in the simple compact form:

$$
\sigma = \mathbb{V} \frac{d^{\alpha} \mathbf{y}(t)}{dt^{\alpha}}, \ 0 < \alpha < 1,\tag{13}
$$

This model provides a generalized viscoelastic response, in the sense that when α ¼ 1 we obtain a Newtonian fluid, and when α ¼ 0 we obtain a Hookean elastic solid. The corresponding mechanical element is intermediate to the spring and dashpot shown in Figure 3 and is thus known as a spring-pot [11, 12]. Note that care must be taken when α ! 1 because of the singularity in Γð1 � αÞ [12].

We can define the fractional Maxwell model (FMM) as a combination of two linear fractional elements (spring-pots) in series. In a series configuration, the stress felt by each spring-pot is the same, that is, <sup>σ</sup> <sup>¼</sup> <sup>V</sup>d<sup>α</sup>γ1ð Þ<sup>t</sup> <sup>=</sup>dt<sup>α</sup> <sup>¼</sup> <sup>G</sup>d<sup>β</sup> <sup>γ</sup>2ð Þ<sup>t</sup> <sup>=</sup>dt<sup>β</sup> , 0 < α, β < 1, and the total deformation is given by the sum of the deformation obtained for each spring-pot, γðÞ¼ t γ t t . The <sup>1</sup>ð Þþ γ2ð Þ FMM can then be written as

$$
\sigma(t) + \frac{\mathbb{V}}{\mathbb{G}} \frac{d^{\alpha-\beta}\sigma(t)}{dt^{\alpha-\beta}} = \mathbb{V} \frac{d^{\alpha}\chi(t)}{dt^{\alpha}},\tag{14}
$$

This model allows a much better fit of rheological data, as shown in [12] but it is not frame invariant. However, following the same procedure employed with the Maxwell and K-BKZ

model, that is, using the derivative of the relaxation function obtained for the Maxwell model as the memory function of the K-BKZ model, one can also use the derivative of the relaxation function of the FMM and insert it in the K-BKZ model, thus, obtaining a frame-invariant constitutive model, that retains all the good fitting properties of the FMM.

The relaxation function of the FMM can be obtained by solving the fractional differential Eq. (14) considering a constant deformation γ ¼ γ0H tð Þ (H tð Þ is the Heaviside function) together with σðt0Þ ¼ σ0, leading to the relaxation modulus G tðÞ¼ σð Þt =γ<sup>0</sup> given by:

$$G(t) = \mathbb{G}t^{-\beta}E\_{\alpha-\beta, 1-\beta} \left(-\frac{\mathbb{G}}{\mathbb{V}}t^{\alpha-\beta}\right),\tag{15}$$

where Ea, <sup>b</sup>ð Þz is the generalized Mittag-Leffler function [7],

$$E\_{\alpha,\beta}(z) = \sum\_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)},\tag{16}$$

and a characteristic measure of the relaxation spectrum described by the two spring-pots in <sup>1</sup><sup>=</sup>ð<sup>α</sup>�<sup>β</sup><sup>Þ</sup> series is <sup>λ</sup> ¼ ðV=G<sup>Þ</sup> .

This leads to the fractional K-BKZ model proposed by Jaishankar and Mckinley [12, 13], with m t <sup>0</sup> ð � t Þ the memory function [2] in Eq. (6) now given by.

$$m(t - t') = \frac{d\mathbb{G}(t - t')}{dt'} = -\mathbb{G}(t - t')^{-1 - \beta} E\_{a - \beta, -\beta} \left( -\frac{\mathbb{G}}{\mathbb{V}}(t - t')^{a - \beta} \right) \tag{17}$$

Note that here the relaxation modulus G tð � t 0 Þ is the one obtained for the FMM. Please see [11–13] for more details. It should be remarked that the Mittag-Leffler function was used in the past by Guy Berry to describe polymeric materials exhibiting Andrade creep [14].

The fractional K-BKZ model is therefore given by:

$$\mathfrak{so}(t) = -\mathbb{G} \int\_0^t (t - t')^{-1-\beta} E\_{a-\beta, -\beta} \left( -\frac{\mathbb{G}}{\mathbb{V}} (t - t')^{a-\beta} \right) h(I\_1, I\_2) \left( \mathbf{C}\_{t'}^{-1} - \mathbf{I} \right) dt', \tag{18}$$

and we need to ensure that the integral converges (see the Foundations of Linear Viscoelastic-<sup>0</sup> �1�<sup>β</sup> ity by Coleman and Noll [15]). The main problem seems to be the term <sup>ð</sup><sup>t</sup> � <sup>t</sup> <sup>Þ</sup> that <sup>Ð</sup><sup>t</sup> <sup>0</sup> ! <sup>t</sup> <sup>0</sup> diverges as <sup>t</sup> , and <sup>ð</sup><sup>t</sup> � <sup>t</sup> <sup>Þ</sup> �1�β dt<sup>0</sup> diverges. Also, the termE<sup>α</sup>�β,�<sup>β</sup>ð Þ … h Ið <sup>1</sup>; I2Þ is finite <sup>0</sup> � C�<sup>1</sup> � <sup>0</sup> <sup>m</sup> <sup>0</sup> �1�<sup>β</sup> ∀t 0 , t<sup>0</sup> ≤ t. Therefore, we must have � I ¼ Oððt � t Þ Þ, m ≥ 1 as t <sup>0</sup> <sup>0</sup> ! t so that ðt � t Þ <sup>t</sup> � <sup>C</sup>�<sup>1</sup> � � <sup>0</sup> <sup>I</sup> <sup>¼</sup> O t ðð � <sup>t</sup> Þ Þ, n <sup>≤</sup> <sup>1</sup> and therefore the integral converges. <sup>n</sup> <sup>0</sup> t

It can be easily shown [1] that <sup>a</sup> Taylor series expansion of <sup>C</sup>�<sup>1</sup> � <sup>I</sup> about <sup>t</sup> <sup>0</sup> <sup>0</sup> <sup>¼</sup> <sup>t</sup> leads to. <sup>t</sup>

$$\mathbf{C}\left(\mathbf{C}\_{t'}^{-1} - \mathbf{I}\right) = -\sum\_{k=1}^{\infty} \frac{\left(t' - t\right)^{k}}{k!} A\_k(t) \tag{19}$$

with Akð Þt the Rivlin-Ericksen tensors. Note that these tensors can be obtained directly from the velocity field without having to find the strain tensor [16]. We may therefore conclude that the integral is convergent, assuming a smooth velocity field is provided/obtained. Note that this does not mean that convergence problems will not arise during numerical calculations.

In Refs. [11, 12, 17], the beneficial fitting qualities of this constitutive model framework are discussed in detail. Here, we are interested in determining to what extent the properties of the Mittag-Leffler function can be used to improve the fitting quality of differential models, and this will be discussed in the next subsection.
