1. Introduction

Since viscoelastic materials are abundant in nature and present in our daily lives (examples are paints, blood, polymers, biomaterials, etc.), it is important to study and understand viscoelastic behavior. Therefore, in this chapter, we further develop the modeling of viscoelasticity making use of fractional calculus tools.

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We start this section with some basic concepts that are needed to derive and understand classical and fractional viscoelastic models. These are trivial concepts such as force, stress, viscosity, Hooke's law of elasticity and also Newton's law of viscosity. Later, we evolve to more complex concepts of viscoelasticity that involve the knowledge of fractional calculus, integral and differential models.

It is well known that a force is any interaction that when unopposed will change the motion of an object/body. Stress is an internal resistance provided by the body itself whenever it is under deformation. Stress is defined as the intensity of internal forces developed in the material. The intensity of any quantity is defined as the ratio of the quantity to the area on which it is acting, leading to: Average Stress = Force/Area. If we want to know the stress in one material point, then we must take the limit of the area to zero. A good example on how stress works is given by imagining a person lying on top of thin layer of ice. When the person is lying down on the ice, the force (weight) divided by the area of the surface of the person in contact with the ice is smaller, when compared to the case when someone is standing up (the weight is the same, but the area in contact with the ice is smaller). Therefore, eventually, the ice will break due to the high internal stresses when the person is standing. Finally, we refer to elasticity as the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. See for example Figure 1 where three springs are stretched. If we remove the weights attached to the springs, the spring would ideally return to its initial/ natural position.

Figure 1(b) also shows an experiment where we observe that the force (mass times gravity) applied to the spring (increasing weight) is proportional to the displacement. This is known as

Figure 1. Schematic of an experiment to obtain the relationship between force and deformation: (a) Experimental setup where three springs are stretched with the use of weights; (b) Graph showing the experimental results obtained from stretching three springs (the force is proportional to the deformation).

Hooke's law (the force F needed to extend or compress a spring by some distance γ ¼ ðx � x0Þ <sup>e</sup> is proportional to that distance, F ¼ kγ ). Note that if we continue to increase the weight, <sup>e</sup> eventually the spring will break. Therefore, Hooke's law is a very good linear approximation of what happens in the real world.

We will now explore the concept of viscosity in fluids. The viscosity of a fluid is a measure of the internal resistance to the rate of deformation:

As an example, imagine that we have a thin film of fluid in between two parallel plates, as shown in Figure 2. The fluid is at rest, and suddenly the upper plate starts moving with constant velocity U. This velocity will be felt at the bottom layer due to diffusion of momentum, and to keep the bottom wall fixed, we must exert a restraining force, that is measured with a force gage or dynamometer attached to that wall. Note that if we take the view of this portion of fluid as infinitesimally thin layers, we observe that each layer will drag the underlying layer due to the action of viscosity (internal resistance). The higher the viscosity, the more force will be required to deform the fluid at a given speed U.

Since the velocity of the thin layer adjacent to the top wall is U and the velocity of the bottom layer is 0, the velocity of each layer (for a Newtonian fluid) is given by u(y) = Uy/h, with y the coordinate shown in Figure 2(a). Figure 2(b) shows the experimental forces measured for different ratios of U/h. We observe that the force is proportional to U/h and U/h = du(y)/dy; therefore, we conclude the following (Newton's law of viscosity):

$$\frac{Force}{Area} = \sigma = \eta \frac{\mathcal{U}}{h} = \eta \frac{du(y)}{dy} \Rightarrow \sigma = \eta \frac{du(y)}{dy} \tag{1}$$

with σ the unidirectional stress and η as the constant of proportionality, known as the Newtonian shear viscosity. Note that du/dy is known as the rate of shear deformation, usually denoted by γ\_.

A good example of something we may see every day and something that verifies Newton's law of viscosity is a dashpot. It is used for example as a door closer to prevent it from slamming shut.

Figure 2. Schematic of an experiment to verify Newton's law of viscosity: (a) Liquid at rest between parallel plates; (b) The top wall is pulled with velocity U and a force meter is used to measure the force exerted on the bottom wall; (c) Experimental results.

### 1.1. Viscoelastic models

The simplest model that considers both viscous and elastic behavior is the linear Maxwell model [1] and can be obtained from a combination in series of a dashpot, σ ¼ ηdγfð Þt =dt, and a spring, σ ¼ Gγ ð Þt (with the subscripts f and e standing for Newtonian fluid and Hookean <sup>e</sup> elastic solid, respectively), as shown in Figure 3.

The total deformation γ is the sum of the deformation obtained from the spring γ and the <sup>e</sup> dashpot γ<sup>f</sup> , and the rate of deformation is given by:

$$\begin{aligned} \frac{d\gamma(t)}{dt} &= \frac{d\gamma\_f(t)}{dt} + \frac{d\gamma\_e(t)}{dt} \\ \Leftrightarrow \frac{d\gamma(t)}{dt} &= \frac{\sigma}{\eta} + \frac{1}{G}\frac{d\sigma}{dt} \\ \Leftrightarrow \underbrace{\sigma + \lambda \frac{d\sigma}{dt}}\_{\text{Maxwell Model}} &= \eta \frac{d\gamma(t)}{dt}, \quad \lambda = \frac{\eta}{G} \end{aligned} \tag{2}$$

The three-dimensional version of this model can be easily obtained by considering appropriate tensors instead of the scalar properties of stress and deformation, leading to the following model:

$$
\sigma + \lambda \frac{d\sigma}{dt} = \eta \frac{d\gamma(t)}{dt} \tag{3}
$$

� � <sup>T</sup> with <sup>σ</sup> the stress tensor, <sup>γ</sup>\_ <sup>¼</sup> <sup>∇</sup><sup>u</sup> þ ð∇u<sup>Þ</sup> the rate of deformation tensor, <sup>u</sup> the velocity vector, λ the relaxation time of the fluid and η the zero shear rate viscosity. This model can be equivalently written in integral form as

$$\mathfrak{o}(t) = \int\_0^t \mathrm{Ge}^{-(t-t')/\lambda} \frac{d\mathfrak{Y}}{dt'} dt',\tag{4}$$

where G ¼ η=λ and it was assumed that the fluid is at rest for t < 0.

Figure 3. Maxwell model.

The Maxwell model is not observer independent (frame invariant) and, therefore, the results obtained with this model may not be correct if large deformations are considered (e.g., we may obtain a viscosity that depends directly on the velocity rather than the velocity gradient, which is not correct, and is unphysical). To solve this problem, new models were proposed in the literature that can deal with this non-invariance problem.

Two well-known examples of frame invariant models are the upper-convected Maxwell <sup>T</sup> (UCM) model given by <sup>σ</sup> <sup>þ</sup> <sup>λ</sup> <sup>σ</sup>^ <sup>¼</sup> <sup>η</sup>γ\_ (with <sup>σ</sup>^ <sup>¼</sup> <sup>∂</sup>σ=∂<sup>t</sup> <sup>þ</sup> <sup>u</sup> � <sup>∇</sup><sup>σ</sup> � ð∇uÞ � <sup>σ</sup> � <sup>σ</sup> � <sup>∇</sup><sup>u</sup> the upper-convected derivative) that can also be written in integral form as

$$\mathbf{u}(t) = \int\_{0}^{t} \frac{\eta}{\lambda^2} e^{-(t-t')/\lambda} \left(\mathbf{C}\_{t'}^{-1} - \mathbf{I}\right) dt' \tag{5}$$

where C�<sup>1</sup> is the Finger strain tensor (a frame-invariant measure of deformation) [1]. The term e�ðt�<sup>t</sup> 0 m t <sup>Þ</sup>=<sup>λ</sup> ð � <sup>t</sup> 0 Þ ¼ <sup>η</sup>=λ<sup>2</sup> is known as the memory function (the derivative of the relaxation modulus Gðt � t 0 Þ). Note that the relaxation modulus can be easily obtained by imposing a step ð Þ=<sup>λ</sup> strain (constant deformation), as shown in Figure <sup>4</sup>, resulting in G tðÞ¼ <sup>σ</sup>=<sup>γ</sup> . <sup>0</sup> <sup>¼</sup> G e� <sup>t</sup>�<sup>t</sup> 0

Other well-known example of a frame-invariant but now nonlinear viscoelastic model is the variation of the K-BKZ [2] model proposed by Wagner, Raible and Meissner [3, 4],

$$\mathfrak{o}(t) = \int\_{0}^{t} m(t - t') h(I\_1, I\_2) \left(\mathbf{C}\_{t'}^{-1} - \mathbf{I}\right) dt',\tag{6}$$

where <sup>C</sup>�<sup>1</sup> is the Finger tensor [1], <sup>I</sup>1, I<sup>2</sup> are the traces of <sup>C</sup>�<sup>1</sup> and <sup>C</sup>, respectively, and h I<sup>ð</sup> <sup>1</sup>; <sup>I</sup>2<sup>Þ</sup> is termed the damping function [5] (note that it is again assumed that the fluid is at rest for t < 0). A large number of damping functions can be found in the literature (see [5]). The term m tð � t 0 Þ was proposed to be of the form:

$$m(t - t') = \frac{a}{\lambda} e^{-(t - t')/\lambda},\tag{7}$$

Figure 4. Step strain of a Maxwell model. The step strain is given by γ ¼ γ0H tð � t0Þ with H tð Þ the Heaviside function, ð Þ=<sup>λ</sup> and the stress relaxation is the solution of <sup>σ</sup> <sup>þ</sup> <sup>λ</sup> <sup>d</sup>σ=dt <sup>¼</sup> <sup>η</sup>dγ0H tð � <sup>t</sup>0Þ=dt with <sup>σ</sup>ðt0Þ ¼ <sup>σ</sup>0, given by <sup>σ</sup> <sup>¼</sup> <sup>σ</sup><sup>0</sup> <sup>e</sup>� <sup>t</sup>�t<sup>0</sup> (σ<sup>0</sup> ¼ Gγ0).

where a and λ are model parameters. Note that the relaxation modulus is the response of the stress to a step in deformation (see Figure 4). It should be remarked that when a ¼ η=λ and h Ið <sup>1</sup>; I2Þ ¼ 1, we recover the integral version of the UCM model.

Different differential models were proposed in the literature along the years, with the aim of improving the modeling of complex viscoelastic materials, and with the aim of achieving the same modeling quality of integral models (by only using differential operators). Note that integral models are non-local (in time) operators that take into account all the past deformation of the fluid while differential models ones describe the material response in terms of the rate of change of stress to the local deformation, thus influencing the fitting quality of the model and the computational effort to numerically solve them (when performing numerical simulations).

More recently, new models have been proposed in the literature that basically take advantage of the generalization of the exponential function appearing in Eqs. (4), (5), and (7), thus allowing a more broad and accurate description of the relaxation of complex fluids (while the commonly used continuum approach describes the fluid as a whole, with only one relaxation, ð Þ unless a Prony series is considered, that is, considering a series of the form P <sup>i</sup> aie� <sup>t</sup>�<sup>t</sup> <sup>0</sup> =λ<sup>i</sup> ). This generalized function is the Mittag-Leffler function that naturally arises when solving problems involving fractional derivatives (more precisely, derivatives of non-integer order). This function will be introduced later in Section 3.
