5 Linear Viscoelasticity

The main goal of this Chapter is to answer the questions:

• What is viscoelastic behavior, and how can it be modeled?

• How do we differentiate between viscoelastic and elastic behavior?

• How do we differentiate between viscoelastic and viscous behavior?

In the lecture, several examples of elastic, viscous, and viscoelastic materials are discussed. For example, we perceive a rubber (elastomer) as a prototypical elastic material, since after loading and subsequent unloading the material attains its original shape (memo: “rubber bands”). Furthermore, we expect that an elastic material shows a stress-strain response that is independent of the applied rate of deformation. The other extreme is a viscous material. If it is loaded and subsequently unloaded, the final shape differs significantly from its original one, and also do the stresses during deformation depend on the rate of deformation, rather on the deformation itself. The third category of materials, viscoelastic, shows a behavior that is between that of elastic and viscous ones. In order to discuss this in detail, we limit ourselves first to the case of linear viscoelasticity.

Also, we restrict our attention in this Chapter to the 1-dimensional case.

For some examples for non-elastic material behavior, see:

• http://www.youtube.com/watch?v=s_3DP3QYqos

5.1Starting point: Hookean solid¶

A purely linear elastic material in 1D can be described by the following stress-strain relation,

which is a Hooke-type relation between the stress $\psigma$ and the shear strain $\gstrn$, with shear modulus $G$. We note in passing that this stress-strain relation can also be expressed as a derivative of a free energy density $A$,

with the help of $A = (G/2) \gstrn^2$. It is manifest that in such (elastic) systems, the stress state depends only on the current strain, irrespective of the deformation history. In other words,

• the current stress state is independent of the rate of deformation, see Figure 5.1;

• the current stress state is independent of whether the material is loaded or unloaded;

• the current stress state is independent of the mode of deformation (oscillatory, steady, ...),

but rather it is only affected by the current state of deformation, by the current value of $\gstrn$.

One can ask the question how much work is done on a mechanical system that is deformed over time. Specifically, if a system is deformed with a certain (time-variable) strain rate $\dot{\gstrn}(t)$ from time $t_1$ until $t_2$, then the work done $\Delta W$ on the system is equal to

If we use the fact that the stress $\psigma$ depends only on the actual strain $\gstrn(t)$, and since $\dot{\gstrn} = d\gstrn/dt$, one can substitute integration variables from time $t$ to strain $\gstrn$, to obtain the work done in terms of a strain-integral,

If we now use the hyperelasticity relation (5.2), one finds

In other words, the work done on the system is given by the free energy difference between final and initial state. This implies particularly, that there is no work done after a complete cycle of deformation, since in this case $\gstrn(t_2)=\gstrn(t_1)$ and therefore also $A(\gstrn(t_2))=A(\gstrn(t_1))$.

5.2Standard measurement protocols¶

We have seen that an ideal elastic solid shows a mechanical behavior that is completely path-independent. If the material is loaded over time, the current stress state depends only on the current strain. This is different for all other than purely elastic materials, and it hence makes sense to define some characteristic measurements, under which the mechanical behavior can be tested. There is a number of standard measurement protocols, namely:

• Step-strain: after a (infinitely) long period of vanishing strain $\gstrn$, the strain is suddenly increased at a certain instant $t_{\rm step}$ from zero to a finite value, and then kept constant again (see Figure 5.2); one is then measuring the stress behavior $\psigma = \psigma(t)$.

• Step-strain rate: after a (infinitely) long period of vanishing strain and strain rate $\dot{\gstrn}$, the strain rate is suddenly increased at a certain instant $t_{\rm step}$ from zero to a finite value, and then kept constant again (see Figure 5.2) in other words, the strain itself increases linearly in time after $t_{\rm step}$; one is then measuring the stress behavior $\psigma = \psigma(t)$.

• (Sinusoidal) Oscillation: the material is in a steady state of oscillatory deformation, where the imposed strain follows a sinusoidal time-behavior with a certain angular frequency (see Figure 5.2); one is then measuring the stress behavior $\psigma = \psigma(t)$.

• Creep: after a (infinitely) long period of vanishing stress $\psigma$ (in other words, no loading), the stress is suddenly increased at a certain instant $t_{\rm step}$ from zero to a finite value, and then kept constant again (see Figure 5.2); in this experiment, one is characterizing the material by measuring the strain behavior $\gstrn = \gstrn(t)$.

In order to get a feeling for these four standard measurement protocols, the reader is encouraged to draw quantitatively the material response for the two extreme cases, namely

• Hookean solid, with “material model” given by (5.1);

• Newtonian fluid, with “material model” given by

  $$\psigma = \eta \ \dot{\gstrn} \ ,$$

  (5.6)

  in other words, the stress is proportional to the instantaneous strain rate, and $\eta$ is called the viscosity.

5.3Origin of viscoelasticity, time scales¶

Imposing deformation on a piece of material brings the material away from its equilibrium state. This can be observed on the macroscopic scale by the emergence of stresses. On the microscopic scale, each constituent particle (atom, monomer, ...) is not anymore in a state of balanced-forces, and hence the deformation results on small scales to unbalanced forces. These unbalanced forces on the microscopic scale and the stresses on the macroscopic scale are different languages to describe one and the same situation: a material is deformed away from its equilibrium shape.

Immediately after imposing the deformation, there are stresses in the material. The question is now what happens with these stresses if the deformation is maintained. Sloppily speaking, materials do not like to be in a stressed state (thermodynamics allows to make more precise statements about this), and if the material has any means to relax the current stress, it will do so. However, how can a material relax stress, while the imposed deformation is maintained? This happens by way of internal relaxation processes. Structural rearrangements occur on the microscopic scale such that the constituent particles evolve towards a configuration/state in which force balance is again respected; that latter state will be observed macroscopically as a state with vanishing stress. Making these necessary structural rearrangements is a dynamic process, and it does thus not come as a surprise that one can associate a characteristic time scale with that process, called relaxation time scale $\taur$.

Based on the existence of a relaxation time scale $\taur$, one can now discuss what happens when, for example, the imposed rate of deformation $\dot{\gstrn}$ is much larger or much smaller than the rate for internal structural relaxation, $\taur^{-1}$:

• Case 1: fast deformation, $\dot{\gstrn} \gg \taur^{-1}$, or equivalently, $\dot{\gstrn} \ \taur \gg 1$: In this case, the imposed (meaning: “enforced”) deformation is so fast, that the internal structural rearrangements are by far not rapid enough to work against force-unbalancing and against the building up of stress. While the materials is deformed more and more, the stresses increase to ever higher values, because the internal structural rearrangements can not keep up with effectively relaxing the structure. In the extreme case, this is what happens in an ideal elastic material, where no internal relaxational processes take place, and the stresses increases as the deformation is increased. In other words, a material with internal time scale $\taur$ behaves close to an ideal elastic solid if it is deformed at a rate so high that $\dot{\gstrn} \ \taur \gg 1$.

• Case 2: slow deformation, $\dot{\gstrn} \ll \taur^{-1}$, or equivalently, $\dot{\gstrn} \ \taur \ll 1$: In this case, the imposed (meaning: “enforced”) deformation is so slow, that at every moment in time the internal structural rearrangements effectively can work against the continuous building up of stress. If the deformation itself (rather than its rate) was kept constant, then the stress would decay to zero after a time span of the order $\taur$. However, since the rate of deformation is constant, and the deformation is continuously increasing in time, the balance between ongoing deformation and relaxation leads to a stationary value of the stress, as is observed for a Newtonian fluid, see (5.6). In other words, a material with internal time scale $\taur$ behaves close to an ideal viscous fluid if it is deformed at a rate so low that $\dot{\gstrn} \ \taur \ll 1$.

It is clear from the above, that a material with internal time scale $\taur$ shows both elastic as well as viscous deformation characteristics if it is deformed at an intermediate rate $\dot{\gstrn} \simeq \taur^{-1}$, in other words $\dot{\gstrn} \ \taur \simeq 1$.

In summary, the characteristic response of the material depends on the ratio of the applied rate of deformation to the internal rate of structural relaxation. For that purpose, it is customary to use the so-called Weissenberg number, defined as

which is a useful dimensionless quantity for discussing viscoelastic phenomena.

Typical time scales do not only occur in classical materials, but they can also be observed in “crowd dynamics”:

5.4Maxwell element¶

5.4.1Basics, definition¶

A useful tool for describing the behavior of materials with both elastic and viscous behavior is to use a combination of the two extremes, which are here summarized again for convenience, in slightly adapted notation:

Particularly, to mimic the behavior discussed in the previous section, we couple the two elements in series (see Figure 5.3), which is then called a Maxwell element. Since they are coupled in series, this implies that the strains and in particular also the strain rates can be added,

With the help of (5.8) and (5.9) this can be cast into the form

which is a $1^{\rm st}$-order differential equation for the stress $\psigma$. The latter can be written is a more intuitive form:

with the characteristic time scale

Since $\eta$ has units of [Pa s] and $G$ has units of [Pa], $\tau$ is indeed a time scale. This time scale plays the role of $\taur$ used in the previous section. However, since in this section we do not make explicit reference to the microscopic structural relaxation processes, we will simply use $\tau$ in relation with the Maxwell element.

It is instructive to discuss the Maxwell model (5.12), before solving it in concrete cases. First, it is apparent that, if the first term on the right hand side was neglected, (5.12) is a simple relaxation equation with relaxation time $\tau$. Second, if all terms are taken into account and considering a constant deformation rate $\dot{\gstrn}$, the stationary state ($\dot{\psigma}=0$) is given by $\psigma = \eta \ \dot{\gstrn}$. Third if the viscosity is infinitely high ($\eta \to \infty$, representative of no structural relaxation), one finds $\dot{\psigma}= G \dot{\gstrn}$, which represents the Hookean limit.

In other words, the serial coupling of a Hookean (elastic) and a Newtonian (viscous) element shows many features discussed above. In order to more precisely quantify the viscoelastic behavior of the Maxwell element defined in (5.12), one needs to perform the standard measurements discussed in Section 5.2: Strep-strain, step-strain rate, (sinusoidal) oscillation, and creep. This will be discussed further below in Section 5.4.3.

5.4.2Relaxation modulus¶

It can be shown^[1] that the solution to the Maxwell model (5.12) can be written in the form

with the relaxation modulus defined by

The explicit solution in the form (5.14) is called a Boltzmann integral. It shows the following features, see also Figure 5.4. First, it shows that the current stress state $\psigma(t)$ is a sum (integral) of effects of the strain rate $\dot{\gstrn}(t')$ in the past ($t' < t$). Second, the strain-rate contribution at an earlier instant $t'$ gets weighted by the factor $G(t-t')$ (which itself is a decreasing function of $t-t'$). Therefore, the further back $t'$, the less does the strain rate $\dot{\gstrn}(t')$ at that moment contribute to the current stress state. This is also in agreement with intuition, saying that there is an effect of fading memory: “the most recent deformation is most relevant, while the earlier deformations are less relevant for the current stress state”. Thirdly, (5.14) also shows that the current stress state depends linearly on the deformation. In other words, doubling the the rate of deformation at all times, leads to a doubling of the stress value. Because of all these features, (5.14) is one of the most important relations in linear viscoelasticity.

5.4.3Standard measurements¶

Earlier, the four standard measurement protocols have been discussed. We leave it to the reader to examine the behavior of the Maxwell model in a step-strain and in a step-strain rate experiment. However, we want to discuss here the cases of creep and sinusoidal oscillation.

Creep: Let us start with the creep experiment. Assume that for $t<0$, the sample is at mechanical equilibrium and no load is applied, $\psigma = 0$. At $t=0$, a stress jump of $\Delta \psigma >0$ is applied, and that value of stress is maintained for $t>0$. How does the material respond? Definitely, at $t<0$, there is vanishing strain, $\gstrn = 0$. To understand what happens at $t=0$, we consider the Maxwell model (5.12), and integrate that differential equation over a small time-interval of duration $\delta$ that includes the instant $t=0$. This integration lads to:

where we have neglected a contribution of the order $\delta$ from the second term on the right hand side of (5.12). This means, that the imposed stress jump results in an immediate strain jump of magnitude $\Delta \gstrn = \Delta \psigma / G$. And what happens at $t>0$? To that end, we again consider the Maxwell model (5.12). Since the stress is held constant for $t>0$, one may neglect the left hand side in (5.12), and one finds the simple relation (with constant value for the stress):

In other words, the strain $\gstrn$ increases linearly in time, at a constant rate given by $\Delta \psigma / \eta$.

Sinusoidal oscillation: In this case, a time-dependent strain is imposed of the form

where $\omega$ denotes the angular frequency. To study the mechanical response of the Maxwell element under this deformation, we start with writing the Boltzmann integral (5.14) in a slightly modified form (transformation of variables from $t'$ to $s = t-t'$),

Using this relation, together with the oscillatory strain (5.18), and employing the general relation $\cos(a-b) = \cos(a)\cos(b)+\sin(a)\sin(b)$, one obtains

where we have defined the so-called storage modulus $G'$ and loss modulus $G''$ by

These are the Fourier-transforms of the relaxation modulus $G(s)$.

According to (5.20), the storage modulus $G'(\omega)$ determines the magnitude of the stress response that is in-phase with the applied deformation (5.18) (proportional to $\sin(\omega \ t)$), and thus describes the elastic-like behavior. In contrast, the loss modulus $G''(\omega)$ determines the magnitude of the stress response that is out-of-phase with the applied deformation (5.18), but in-phase with the applied rate (proportional to $\cos(\omega \ t)$), and thus describes the viscous-like behavior; this explains why $G''$ is called the “loss” modulus.

There are then two ways to quantify the material behavior. Either, by knowledge of the entire relaxation modulus as a function of time, $G(t-t')$, or my measuring in oscillatory measurements the two moduli $G'(\omega)$ and $G''(\omega)$. However, it must be pointed out that the $G'(\omega)$ and $G''(\omega)$ must be known not only at one specific frequency, but rather (ideally) at all frequencies, or at least in a wide frequency range.

In order to make the notion of in-phase and out-of-phase with the applied deformation more clear, the relation (5.20) is often rewritten in the form^[2]

with the definitions

where the quantity $\delta(\omega)$ is called the phase shift, i.e. the lag of the stress response with respect to the applied deformation (5.18) (see Figure 5.5). It is thus the ratio of loss to storage modulus that determines how much the stress response is in-phase with the applied deformation.

While the relations (5.21) and (5.22) are the general expressions in terms of the relaxation modulus $G(s)$, it can be shown with the aid of the special relaxation modulus (5.15), that the storage and loss moduli for the Maxwell element are given by the relations (see Figure 5.6)

Consequently, the phase shift of the Maxwell element becomes

It is rather illuminating to study for (5.28) the limiting cases of very fast or very slow deformation, and check whether they agree with the intuitive discussion presented in Section 5.3. This is left as an exercise for the reader.

The notion of $G''(\omega)$ as a quantity related with “loss” (in other words: dissipation) can also be seen by the following argument. Using the relation (5.3) for the work done on the system, and applying that to a system described by the constitutive relation (5.20) with applied deformation (5.18), one obtains for the work done in one full cycle

Therefore, the work done in one full cycle depends only on $G''$, but is independent of $G'$.

5.5Several Maxwell elements in parallel¶

The relaxation modulus (5.15) is an exponential decay with relaxation time $\tau$. Particularly, the Maxwell element describes a fluid, since for example after a step strain, the stress decays to zero after sufficiently long time. When studying common viscoelastic fluids, they also show that stress relaxation behavior, however, not with a single relaxation process, but with a superposition of relaxation processes, with respective time scales $\tau_i$. The observed behavior can be well described by a parallel arrangement of Maxwell elements, which are then called different Maxwell modes, and thus the relaxation modulus of such a viscoelastic fluid is not given by (5.15), but rather by its generalization

with the mode-specific relaxation modulus

with $\tau_i$ the relaxation time of mode $i$. Of course, the presence of different modes has ramifications for the loss and storage moduli, $G'$ (5.21) and $G''$ (5.22). In the multimode case, they are not anymore given by (5.26) and (5.27), but rather by their multimode counterparts

In order to identify the various parameters of the modes, sinusoidal oscillation measurements are preferred, for example over step-strain (meaning: stress relaxation). In stress relaxation, one should split up a decaying function into a superposition of various exponential decays. This is for practical purposes cumbersome. Much easier and more accessible are sinusoidal oscillation measurements: The storage modulus $G'(\omega)$ transitions from one plateau value to another at the characteristic frequencies $\omega = \tau^{-1}_i$, and even more prominent are the peaks of the loss modulus $G''(\omega)$ at the same characteristic frequencies $\omega = \tau^{-1}_i$ (see Figure 5.7). It is significantly easier to discriminate superposed peaks in $G''(\omega)$, rather than to discriminate superposed exponential decays in $G(t-t')$. This is the reason why usually oscillatory measurements are used to characterize a viscoelastic fluid.

5.6Kelvin-Voigt element¶

As an alternative to the Maxwell element, in the literature also the so-called Kelvin-Voigt element is used. It consists from the same elements as he Maxwell element, a Hookean and a Newtonian element, however, in the Kelvin-Voigt element these elements are arranged in parallel (see Figure 5.8). The Kelvin-Voigt element has a rather undesirable feature for most practical mechanical engineering purposes. Namely, in fast loading (high rate such as in shock; step-strain experiment), the Newton element gets completely stiff, meaning that the Kelvin-Voigt element is not deformed at all. For this reason, we do not discuss the Kelvin-Voigt element any further in this lecture.

5.7Standard Solid element¶

For modeling purposes, or for the stability of numerical implementations, the Maxwell element is extended to include a Hookean element in parallel to the Maxwell element (see Figure 5.9). While within the Maxwell element the strain rates are to be added (see (5.10)), the parallel coupling of the Maxwell element with the additional Hookean element requires adding the corresponding stresses. This new composed element is called Standard Solid element, or Standard Solid model.

Most common is the case, where the elastic modulus $G_1$ in the Maxwell element is much larger than the elastic modulus $G_2$ of the additional Hookean element, $G_1 \gg G_2$. In this case, the mechanical response in the high-deformation rate regime is dominated by the Maxwell element (since the Newtonian element is rather stiff and thus the load is effectively carried by the $G_1$-spring. However, at very low deformation rates, only rather small loads are built up over the Maxwell element (Newtonian element basically flows), and so the additional parallel $G_2$-elastic element carries the load.

Rather than going into more details here, we leave this to the interested reader. For example, one could examine the behavior of the Standard Solid element in the four standard deformation protocols.

For completeness, it is mentioned that if a spring is combined in parallel not only with a single Maxwell element (as in Figure 5.9), but rather with several Maxwell elements in parallel, then this model is called the Generalized Maxwell element, or Generalized Maxwell model.

5.8Check your understanding¶