6 Viscoplasticity

In this Chapter, we extend our considerations of the Maxwell element in the previous Chapter. Particularly, the purpose of this Chapter is to discuss the following issues:

• How does the material response (for example of a Maxwell element) change, if the viscosity depends strongly on the current stress?

• How does the material response (for example of a Maxwell element) depend on the temperature?

In other words, we have discussed in the previous Chapter linear viscoelasticity. In this Chapter, we seek nonlinear modifications of that approach, particularly as far as stress- and temperature-dependencies are concerned.

Looking at the Maxwell element discussed in the previous Chapter, there are in principle two spots where a stress-dependence could be introduced: First, the Hookean element could be made nonlinear, meaning that the modulus $G$ becomes stress-dependent. In this Chapter we concentrate instead on the second possibility, which is to make the viscosity stress-dependent. It will be discussed that this does not only lead to quantitative changes in the material behavior, rather it amounts to qualitatively different mechanical response. By explaining the physical origins of this stress-dependence of the viscosity, it also becomes clear that the mechanical response of a viscoelastic material must depend substantially on temperature, the second main topic in this Chapter.

6.1Kinetics of structural rearrangements¶

In Section 5.3, “Origin of viscoelasticity, time scales”, we have discussed the relevance of structural rearrangements on the microscopic scale to relax stresses that arise due to macroscopically imposed deformation. The conclusion was that viscous-like behavior on the macroscopic scale has its origin in small-scale kinetics of the constituent particles (atoms, molecules, ...).

We now look in more detail into this small-scale kinetics. A wide-spread working hypothesis all across natural sciences and engineering disciplines is that (rate-limiting, barrier-crossing) kinetics can be described by an Arrhenius law,

with $k$ the kinetic rate, $k_0$ the kinetic-rate prefactor, $U$ the energy barrier, $\kB$ the Boltzmann constant, and $T$ the absolute temperature. In this concept, one is obviously not interested in the fast vibrations of atoms and the like, rather one looks for transitions that lead to significant changes in the structure, something that we as mechanics people would identify with an inelastic/plastic event.

The above relation shows that such structural rearrangements depend on the ratio of the energy barrier $U$ to the thermal energy $\kB T$. This implies that the kinetics can be accelerated (slowed down) upon increasing (decreasing) the temperature. However, there is also other means to accelerate the kinetics of structural transitions. Eyring (Krausz, A., Eyring, H., 1975. “Deformation Kinetics”. Wiley-Interscience, London) has put forward the following idea of stress-assisted kinetics:

Imagine that the microstructure can change in two possible “directions”. Then, in equilibrium, the rates to transition in either of the two directions, denoted by $k_>$ and $k_<$, are equal, namely they are both given by (6.1). However, when a load is applied in the form of a stress, there is a bias to transition preferentially in one of the two directions. In terms of the Arrhenius relation (6.1) this effect can be captured by noting that the (free) energy barrier is modified by the applied stress. More specifically, Eyring stated that the energy barrier is modified in a linear manner in terms of stress,

which implies that the energy barrier in the “forward” direction is decreased, while the energy barrier in the “backward” direction is increased (see Figure 6.1). The material parameter $v$ (units: [m$^3$]), $v > 0$, is introduced for dimensional reasons, and is called the activation volume. This leads to an imbalance in forward and backward rates that reads as follows:

If one then identifies the imbalance in rates, $k_> - k_<$, with the macroscopically observed plastic (=unrecoverable) strain rate $\psr$, then one can write, as far as the stress-dependence is concerned,

That plastic strain rate in turn can be used to define the viscosity in the following way:

where the prefactor $\eta_0$ has been introduced for convenience. Particularly, one finds (since $\sinh(x) \simeq x$ for $x \ll 1$) that $\eta(\psigma) \simeq \eta_0$ for vanishingly small loads, $v \, \psigma \ll \kB T$. On the other hand, the viscosity $\eta(\psigma)$ drops rapidly below $\eta_0$ for large loads, $v \, \psigma \gg \kB T$ (see Figure 6.2). The viscosity function defined in (6.5) is called the Eyring viscosity.

The Eyring viscosity (6.5) is particularly useful to describe solid polymers, and it has an intuitive origin. On the other hand, for example in metals, other relations are used (see also lecture slides). Most common in the field of metal plasticity is to work with a plastic strain rate that depends through a power-law on the applied stress, $\psr \propto \psigma^m$. The viscosity is then defined correspondingly as $\eta = \psigma/\psr \propto \psigma^{1-m}$. Usually, one has for the power-law exponent $m \gg 1$, leading, as in the Eyring case, to a rather strong nonlinearity. In the case of metal plasticity, the stress-dependence of the viscosity originates from the collective effects in the dynamics of dislocations, the carriers of plastic deformation in crystals.

6.2Viscoplastic model¶

In the previous section, possible origins for the stress-dependence of the viscosity have been discussed. Using the Maxwell model (5.12) discussed in the case of linear viscoelasticity, taking the stress-dependence into account leads to

with the now stress-dependent characteristic time scale

One viewpoint of (6.6) is to assume that the relation $\eta=\eta(\psigma)$ is given. On the other hand, one can also take the viewpoint that the material at hand be described by a viscoplastic model of the form (6.6), and the question is then how the relation $\eta=\eta(\psigma)$ can be determined from experiments. Of course, the standard experiments come to mind: step-strain, step-strain rate, (sinusoidal) oscillation, and creep. For step-strain and creep, the reader may follow procedures similar as in Linear Viscoelasticity. As far as sinusoidal oscillation is concerned, we recall from Linear Viscoelasticity that the analysis there profited essentially from the fact that the model was linear^[1]. One can already anticipate that the nonlinearity in (6.6) due to the stress dependence of the relaxation time will complicate the analysis of oscillatory response significantly. Therefore, we concentrate here on the case of a step-strain rate experiment.

In the step-strain rate experiment, let us consider only the situation in which the stress $\psigma$ has come to a stationary state, $\psigma_{\rm steady}$. According to (6.6) we have have the following condition:

This equation can be used to examine the the dependence of the steady “yield” stress $\psigma_{\rm steady}$ as a function of the applied (constant) strain rate $\dot{\gstrn}$. Obviously, the found relation depends on the viscosity function $\eta(\psigma)$. In turn, this implies that measuring the relation $\psigma_{\rm steady} = \psigma_{\rm steady}(\dot{\gstrn})$ allows to access the relation $\eta(\psigma)$. For example, for the case of the Eyring viscosity (6.5), the steady-state condition (6.8) assumes the form

Using $v \ \psigma_{\rm steady} \gg \kB T$ (which can be justified a posteriori by analyzing experimental data), one can use the approximation $\sinh(x) \simeq \exp(x)/2$, which leads to

The last relation can be inverted readily to find the dependence of the steady-state yield stress $\psigma_{\rm steady}$ on the applied constant strain rate $\dot{\gstrn}$,

In particular, we find the following relation that allows the experimental measurement of the activation volume,

In other words, when plotting the steady-state yield stress as a function of the natural logarithm of the applied strain rate, the slope in that curve is directly related to the activation volume. And, indeed, this is how in practice the activation volume for a material is determined experimentally (see Figure 6.3).

6.3Terminology¶

In this section, we differentiate between different forms of non-elastic behavior, and clarify particularly the difference between three terms, namely: viscoelasticity, elasto-plasticity, and elasto-viscoplasticity.

• Viscoelasticity

  • both elastic and viscous behavior (for example: Maxwell element)

  • there can also be non-linear effects (for example: stress-dependent elastic modulus or stress-dependent viscosity)

  • typical for polymeric materials

• Elasto-plasticity

  • there is a critical yield stress, below which the material behaves purely elastic; plastic flow is only possible above that yield stress

  • typical for metals at low temperature

• Elasto-viscoplasticity

  • primarily elastic behavior at low stresses; there is still a small amount of viscous effects at small stresses (for example, see the Eyring model), in other words: even at low stress values there is a small amount of flow

  • primarily plastic flow at high stresses

  • elasto-viscoplasticity is an example of strongly non-linear viscoelastic behavior

  • typical for solid polymers and metals at high temperatures

6.4Effect of temperature: Time-temperature superposition principle¶

At finite absolute temperature, there is always some degree of mobility in the system. For example, one can show that for a material at finite temperature $T$, the average kinetic energy of a particle (atom) is proportional to $T$. As a consequence, there is increased (molecular) mobility in the system as the temperature $T$ is increased. As an illustration, one can consider the melting of ice:

What does the increased temperature mean for macroscopically measurable quantities that relate to the dynamics? Well: as the temperature is increased, and as the particles move more rapidly, the time scale for structural rearrangements decreases, as does the viscosity of the system (see (5.13) for the relation between the time scale $\tau$ and the viscosity $\eta$).

An illustrative example is a dense system of polymer chains that are entangled (like a pot of intertwined spaghetti). In this system, the time for disentanglement is the longest time scale in the system. Let us first study the material behavior of this system at a fixed temperature, with increasing frequency (see Figure 6.4). If the frequency is very high, and the rate of deformation thus very fast, the entangled chains do not have the time to disentangle in order to relax the rapidly building-up stress, and the system therefore reacts as if the entanglements were permanently fixed (chemically cross-linked). Decreasing the frequency to very small values leads to a deformation rate that is so small that the chains do have the time to rearrange the entanglement-structure to relax the stresses, and therefore the system behaves fluid-like. While this is an argumentation along the lines of changing frequency at a fixed temperature, one can also look at the change in material response at a fixed frequency while changing the temperature $T$. In analogy to the arguments just mentioned, one can anticipate the following scenario: At a fixed frequency and very low temperature, the mobility in the system is rather small with a characteristic rate of rearrangement much smaller than the applied frequency, and thus the material behaves as a chemically cross-linked system. If, however, the temperature is increased drastically, the mobility is so much increased that even entanglement-rearrangements are involved in relaxing the imposed deformation. In short, increasing the temperature (at fixed frequency) shows the same effect as decreasing the frequency (at fixed temperature). The questions is now: How can we rationalize that finding, and make use of it in viscoelastic and elasto-viscoplastic models?

To study the effect of temperature on the rate-dependent behavior of materials, it is useful to define the time-temperature shift factor for a certain relaxation process $j$,

If the material behavior is known at a certain temperature $T_0$, and a corresponding model has been set up, then one can also apply that model to the behavior at another temperature $T$, by replacing all time scales $\tau_j(T_0)$ in the model by $\tau_j(T)$. Notably, one needs to make that replacement for all processes $j$. This implies that one in principle would need to know the time-temperature shift factors $a_{T_0,j}(T)$ for all processes. This is in general a formidable task, meaning that the experimental determination of the different shift factors is difficult. However, it turns out that for many polymeric materials the time-temperature shift factor is the same for all (measurable) processes (meaning: the time-temperature shift factor is independent of $j$),

in which case the material is called to be thermorheologically simple. The (experimentally supported) assumption that the effect of temperature can be included in a material description by an appropriate rescaling of the time-scale is called the time-temperature superposition principle. An example for the time-temperature shift factor is shown in Figure 6.5.

We leave it as an exercise for the reader to examine the effect of lowering the temperature on the behavior of a Maxwell element in the four standard measurements (step-strain, step-strain rate, (sinusoidal) oscillation, and creep).

The time-temperature superposition principle of thermorheologically simple materials is very beneficial for practical measurements. Imagine, for example, that you want to measure in a step-strain experiment the relaxation modulus $G(t, T_0)$, at a certain temperature $T_0$. Most probably, the experimental set-up you are using has a limited time-resolution: On the one hand, there is a lower bound on the resolvable dynamics $t_{\rm min}$, in the sense that the mechanics of the apparatus has certain inertia, and thus very small time-scales and fast deformations are inaccessible. On the other hand, there is an upper bound on the resolvable dynamics, $t_{\rm max}$, for example because you can not keep the temperature constant at $T_0$ over a long enough period. In other words, you can only measure the relaxation modulus $G(t,T_0)$ for times $t_{\rm min} < t < t_{\rm max}$. For thermorheologically simple materials, one can still determine the relaxation modulus $G(t, T_0)$ for times $t$ outside the range $t_{\rm min} < t < t_{\rm max}$, as described in the following.

Imagine that the relaxation modulus $G(t, T_0)$ is given by the relation (5.15), where for simplicity we restrict our attention to a single mode only. In this expression, one observes that the relaxation modulus is a function of the ratio $t/\tau$ with $\tau$ the relaxation time, and no other time-dependence occurs. Particularly, one can write the relaxation modulus in the form

for a certain function $f$. Using the time-temperature shift factor (6.13), one can rewrite this expression into

In other words, the value of the relaxation modulus at a certain time $t$ measured at temperature $T_0$ is equal to the relaxation modulus at time $t \ a_{T_0}(T)$ measured at temperature $T$. What does that mean in practice? Suppose that you want to determine the relaxation modulus at a very short time $t < t_{\rm min}$ (which is outside the experimentally accessible window) at temperature $T_0$. So, you are interested in fast processes. These fast processes are slowed down upon decreasing the temperature to $T$ (therefore: $T < T_0$), which is described by (6.13) with $a_{T_0}(T) > 1$ for $T < T_0$. The relation (6.16) then tells you that, instead of measuring the relaxation modulus at time $t$ and temperature $T_0$, you can rather measure the relaxation modulus at a lower temperature $T < T_0$ but at a longer time $t \ a_{T_0}(T)$ (note that $t \ a_{T_0}(T) > t$). The idea is thus to go to so low temperature $T$ that this “longer time” falls within the possible measurement window, $t \ a_{T_0}(T) > t_{\rm min}$. After performing this measurement at $T$, one can “shift” the result according to (6.16), in order to obtain eventually the time-dependence of $G(t, T_0)$ over a time-range much wider than $t_{\rm min} < t < t_{\rm max}$. This procedure is demonstrated in Figure 6.6.

While the above procedure (changing temperature- and time-window) was explained on the example of a single-mode model, the question arises how to do that for a Maxwell model with multiple modes, in other words with multiple relaxation times. This is exactly the point when it is useful to know whether a material is thermorheologically simple. For this latter class of materials, one can use the same shift factor for all processes. As a consequence, the relation outlined in (6.16) can be used in exactly the same way also for multimode materials, for example as described by (5.30) and (5.31) .