In all FE simulations we run, we always have to specify the bulk modulus (or Poisson’s ratio) of all material models. In this article I will try to answer the questions: Is it important to know the bulk modulus? And does the value we specify make a big difference?
I will investigate this in a 3 different ways.
1. Uniaxial Tension of a Thermoplastic Material
We cannot easily change the bulk modulus (\(\kappa\)) of real materials, so it is easiest to perform our investigations using “virtual materials”. That is, I will use an accurate viscoplastic material model for thermoplastic material, here I will use the PolyUMod Three Network (TN) model, and examine the response in uniaxial tension for different moduli values. The following figure shows the results from MCalibration, which is a perfect tool for this type of investigation. The figure shows that when I change \(\kappa\) from 500 MPa all the way to 2000 MPa, there is virtually no change in the stress-strain response.
2. Uniaxial Tension of a Rubber
For completeness, I will repeat this test using a non-linear viscoelastic material model suitable for rubbers (the Bergstrom-Boyce model). In this case I varied \(\kappa\) between 100 MPa and 2000 MPa. These values correspond to Poisson’s ratios between 0.48 and 0.499. The figure below shows that the stress-strain response is not changing in any significant way for this range of bulk modulus values.
3. Compression of an O-ring
The first 2 case studies showed that the bulk modulus does not significantly influence the stress-strain behavior in uniaxial tension. But what happens in a more realistic multiaxial test case? Here I will investigate the behavior of an o-ring that is compressed in a groove with a gap. I created a simple axisymmetric FE model using Abaqus. The mesh and boundary conditions are shown below. I simply applied a pressure on the bottom half of the o-ring in order to compress it. This may not be the most accurate method, but it is sufficient for our case study. I then assign the Bergstrom-Boyce material model to the o-ring.
The two figures below show contours of the min principal true strain for a case with \(\kappa=400\) MPa, and a case with a \(\kappa=2000\) MPa. The figure also summarizes the max tensile strains and the max Mises stress for the two cases. It is quite interesting to see that both FE simulations give essentially identical results for the two bulk modulus values. Apparently the bulk modulus does not strongly influence the response even in this highly confined geometry.
How to Experimentally Measure the Bulk Modulus
I recommend the following methods if you need to experimentally measure the bulk modulus of a polymer.
- Digital Image Correlation (DIC) of a tension test
In this case you measure both the axial and transverse strains, which will allow you to determine the Poisson’s ratio, from which you can determine \(\kappa\). This works, but sometimes gives inaccurate results due to DIC noise.
- Pressure – Volume – Temperature (PVT) testing
There are specialized test equipment that allow you to measure the specific volume as a function of pressure, from which you can calculate the bulk modulus.
- Confined Compression Testing
In this method you compress a cylindrical specimen in a hollow cylinder while you measure the pressure as a function of applied volume. This test is described in more detail in Section 2.2.9 of my “Mechanics of Solid Polymers” book.