The Math Behind Stress Relaxation and Creep of Polymers

Introduction

In two recent articles I discussed the experimentally observed stress relaxation response of thermoplastics and rubbers. I also focused on how to extrapolate existing short-term experimental stress relaxation data to longer times. In this article I will focus on some basic equations that control the stress relaxation and creep response of a general polymer material models. I will demonstrate the response of both linear and non-linear viscoelasticity, and I will do this first by closed-form derivations, and then using MCalibration to perform the calculations.

Maxwell Model - Stress Relaxation

A Maxwell rheological model is a simple linear viscoelastic model consisting of a linear spring in series with a linear dashpot. The spring has a stiffness of \(E\), and the dashpot has a linear viscosity of \(\eta\). The governing equations for a linear viscoelastic Maxwell model are:
\( \begin{cases}
\sigma &= \varepsilon_1 E\\
\dot{\varepsilon}_{2} &= \sigma / \eta\\
\varepsilon &=\varepsilon_1 + \varepsilon_2
\end{cases}\)
If a strain jump is applied at \(t=0\), then it is easy to show that the stress relaxation relaxation response becomes: \(\sigma(t)=E\,\varepsilon_0\, e^{-Et/\eta}\). Note that if you have not performed this type of calculation before, then I recommend that you spend a few minutes to derive this equation yourself. It is a great exercise! If we pick E=1 MPa, \(\varepsilon_0=0.1\), and \(\eta=1\) MPa s, then the stress relaxation response becomes the following (plotted using Julia). Note that the stress relaxation response becomes a straight line if the stress axis is logarithmic. This is different than the plots that I presented for rubbers and thermoplastics!

				
					using Plots
x = range(0.01, 10, length=100)
y = exp.(-x)
plot(x,y, yaxis=:log)
plot!(xlabel="Time", ylabel="Stress", size=(600,400))
plot!(legend=false, framestyle=:box)
savefig("maxwell_relax.png")
				
			

Maxwell Model - Creep

The creep response of a linear viscoelastic Maxwell model is even easier to derive. The creep rate of the dashpot is given by \( \dot{\varepsilon}_2 = \sigma_0 / \eta \), so the total creep strain as a function of time becomes:

\( \varepsilon(t) = \sigma_0 / E + \sigma_0 t/\eta\).

That is, the creep strain is increasing linearly with time. The following figure shows the predictions for the case when E=1 MPa, \(\sigma_0\)=0.1 MPa, and \(\eta=1.0\) MPa s.

				
					using Plots
x = range(0.01, 10, length=100)
y = 0.1 .+ 0.1*x
plot(x,y)
plot!(xlabel="Time", ylabel="Strain", size=(600,400))
plot!(legend=false, framestyle=:box)
savefig("maxwell_creep.png")
				
			

Two Network Rheological Model - Stress Relaxation

The mechanical response of almost no solid polymer can be accurately represented using a single Maxwell model. Typically, much better predictions can be obtained by switching to a two network model, where the first network (A) gives the equilibrium response, and the second network (B) gives the viscoelastic relaxation response. This is similar to a Prony series linear viscoelastic model where the response improves when you use more than one term. The stress relaxation response for the case when the springs are linear elastic and the dashpot has constant viscosity is trivial to obtain (since the networks are in parallel):
\( \sigma(t) = E_A \varepsilon_0 + E_B \varepsilon_0 e^{-E_0t / \eta} \).

The stress relaxation response of this model does NOT become linear using any combination of x-axis or y-axis log scales. The most useful plot in this case is often obtained using a log axis for time. The following figures show a few different plots of the same relaxation data.

Two Network Rheological Model - Creep

The creep response of two parallel networks with linear springs and dashpot can also be derived in closed-form. If I define \( \hat{E} = E_A E_B / (E_A + E_B) \), then the creep strain becomes the following for a step in stress:
\( \varepsilon(t) = \displaystyle\frac{\sigma_0}{E_A} \left[ 1 – \frac{\hat{E}}{E_A} e^{-\hat{E}t/\eta} \right] \)

The creep response of this model does not become linear using any combination of x-axis or y-axis log scales. The most useful plot in this case is often obtained using a log axis for time. The following figures show a few different plots of the same relaxation data.

Three Network Non-Linear Viscoelastic Model (TNV)

For more realistic (and accurate!) material models it is not always possible to derive closed-form stress relaxation or creep equations. In cases like this it is fortunately possible to quickly determine the material model response using MCalibration. As an example, I created a virtual creep data file and selected a 3 network PolyUMod TNV model with the following parameters: C10=1 MPa (all 3 networks), tauHat2=0.5 MPa,  m2=6, tauHat3=0.01 MPa, m3=6. The predicted stress relaxation response for this model is qualitatively similar to the two-network models discussed above, but due to the non-linear model components, the relaxation response can be tailed to experimental data more accurately than a linear viscoelastic model.

The following figure shows the predicted creep response of the TNV model.

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