Volumetric Relaxation in a Prony Series

Introduction

A linear viscoelastic material model is defined using an elastic response together with a Prony series relaxation. Here’s an example using Ansys APDL:

				
					
TB, HYPER, matid, 1, 2, NEO
TBDATA, 1, [mu]
TBDATA, 2, [D]
TB, PRONY, matid, 1, [N], SHEAR
TBDATA, 1, [alphaG1]
TBDATA, 2, [tau1]
...
TB, PRONY, matid, 1, [N], BULK
TBDATA, 1, [alphaK1]
TBDATA, 2, [tau1]
...
				
			

A linear viscoelastic material model can also be defined using the following Abaqus commands:

				
					*Hyperelastic, Neo Hook, moduli=instantaneous
[C10], [D1]
*Viscoelastic, time=prony
[g1], [ki], [taui]
...
				
			

In both of these cases the [alphaK] and [k] parameters specify how much the bulk modulus changes as a function of the relaxation time. In most cases the bulk modulus relaxation is not experimentally measured, and all relaxation is simply assumed to be due to shear deformations. This has many similarities to basic hyperelasticity where the material is often assumed to be almost incompressible and the shear terms are obtained from uniaxial stress-strain data. This works since in practice the bulk modulus is not very important for actual stress-strain predictions.

In this article I will illustrate specific cases where you  should consider include volumetric relaxation in your linear viscoelastic Prony series. It turns out that it is easy to do this with MCalibration!

Experimental Data

Figures 1 and 2 show the experimental data that I used for the study. The DMA experiments were performed in uniaxial tension, and the Time-Temperature-Superposition (TTS) master curve was generated from data at different temperatures.

DMA Storage Modulus

Figure 1. Experimental storage modulus as a function of frequency.

DMA Loss Modulus

Figure 2. Experimental loss modulus as a function of frequency.

Basic Material Model Calibration

Figure 3 shows the results from MCalibration’s instantaneous linear viscoelastic parameter selection. The parameters that are selected by MCalibration are based on the assumption that all viscoelastic relaxation is shear driven. The predictions are obviously very accurate, but what are the consequences from the assumption that all relaxation is shear driven?

Ansys LVE calibration.

Figure 3. Predicted storage and loss modulus response.

The calibrated material model in this example is based on Neo-Hookean hyperelasticity together with a 30-term Prony series. In MCalibration I can quickly examine the stress-strain predictions from the calibrated material model in, for example, uniaxial tension tests at a constant strain rate (using a “Virtual Experiment”). In Figure 4 below I selected 3 widely different strain rates (1e-6/s, 1e6/s, and 1e20/s). As expected, the stress-strain response becomes strongly strain-rate dependent. The Poisson’s ratio of the calibrated material model depends on the hyperelastic parameter d and the amount of volumetric relaxation (which here is set to zero). The figure shows that the Poisson’s ration is between 0.47 and 0.499 depending on the strain rate. 

volumetric relaxation - stress

Figure 4. Predicted stress-strain and Poisson’s ratio behavior.

Second Material Model

As a second example, I kept all parameters the same except that I increased the d parameter by 10X to 0.01/MPa. In this case stress-strain predictions at the highest strain rate becomes a bit softer, but more interestingly, in this case the predicted Poisson’s ratio becomes highly dependent on the applied strain rate. It is seems unlikely that the Poisson’s ratio is about 0.499 at slow rates, and less than 0.3 at high rates. More experimental data would be needed to determine if that response is good or crazy! My bet is on crazy…

volumetric relaxation - stress2

Figure 5. Predicted stress-strain and Poisson’s ratio response for a material with lower bulk modulus.

A Third Material Model with a Constant Poisson's Ratio

It is easy to modify the material model so that the predicted Poisson’s ratio becomes basically independent of the applied strain rate. All we need to do is to make the bulk modulus relax the same way as the shear modulus. Figure 6 shows a screenshot of MCalibration where I made each rel_k value equal to the corresponding real_g value, and I also made sum_k equal to sum_g. The predictions from this material model is shown in Figure 7. As before, the magnitude of the Poisson’s ratio is mainly determined by the d value, but the strain-rate dependence of the Poisson’s ratio is in this case almost gone.

LVE Material Parameters

Figure 6. Material parameters for the modified material model.

Figure 7. Predicted stress-strain and Poisson’s ratio for the modified material model.

Summary

  • In many (perhaps most) cases it is sufficient to simply set the volumetric relaxation to be 0, and select the compressibility parameter d in order to get the target Poisson’s ratio.
  • If the storage modulus is changing A LOT as a function of frequency (i.e. if sum_g is almost 1) then is often better to set the volumetric relaxation to be the same as the shear relaxation in order to get a more realistic Poisson’s ratio at all strain rates.
  • As always, it is better to experimentally measure the bulk modulus relaxation response. This can be tricky to do in practice. I plan to write an article about that in the future.
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