How to setup Triaxial Loading for a Voided Media

I got a direct email from a student about the question of assessing features of a ductile response in which the user wants to ensure constant triaxiality with a unit cell analysis. The student provided me with two journal articles with the most recent of them given below:

Tekoglu, C., 2014. Representative volume element calculations under constant stress triaxiality, lode parameter, and shear ratio. International Journal of Solids and Structures, 51(25-26), pp.4544-4553.

Video showing setup of Triaxial Loading

I studied the paper and really enjoyed it. It inspired me to formulate a clearer way to help the student. That led to the video below which I published this week simply titled: How to apply Triaxial Loading on 3D RVEs in ABAQUS (voided material). In the 2 days since it was published, it has gathered already 286 views. This is quite good in comparison to the average videos on the challenge. If you have not watched the video, then here is it:

Hallmarks of the Triaxiality challenge

There are certain things that we learn from the video above which I have termed the Triaxiality Challenge. This is because everyone interested in working with triaxial loading on RVEs must address these features. I believe these were the features that the student who contacted me was struggling with. They are the hallmarks of the triaxiality challenge:

1.0 Virtual Domain generation

Every RVE study must always address the challenge of recreating the RVE of the problem. In most cases, the RVE can be quite simple in which case straightforward to recreate. However, many real life analysis will involve an RVE that is challenging to create. This is precisely the reason for isolating an RVE as against modelling the whole architecture. In the referenced paper above, they studied a single central void within a cubic RVE. I decided to extend the challenge to more realistic systems by exploring multiple voids with random distribution. I had to resort to a bespoke software developed by my company - CM Videos Ltd - designed precisely for such problems. This code is called MontCarlGen3D version 1. It is not yet available for public use but if you are interested in getting informed on when it becomes available, then join the mailing list of the software by signing up here.

2.0 What problem are you solving?

It is important to also understand the problem you are trying to solve with the triaxiality challenge. In the video above, I identified application areas where triaxial loading can come in handy for example: (a) exploring the coalescence of voids (b) dilation and (c) hydrostatic studies and well as (d) characterization of features of ductile deformation as: stress triaxiality, lode parameter and shear stress ratios. The last application area was the basis of the referenced paper. It was also the basis upon which my video was made although I did not get to the end of actually determining these parameters. If you are interested then do consider the paper. One glaring miss from my video is the absence of inplane shear loads. This was on purpose as I already have a different video where I showed how to impose pure shear loading on RVEs which you can watch here:

3.0 Unique application of Boundary Conditions (BC)

Since this is a triaxiality challenge, it is crucial that the user finds a simple and effective way to apply the boundary conditions that will impose the triaxial loading at every time step during the simulation. I did this in my video by combining a few set of unique features namely:

• Using nodal surface sets to apply Dirichlet boundary conditions so that three orthogonal planes (X-, Y- and Z-surfaces were held securely with a roller boundary condition).
• Applying displacement loads on three orthogonally positioned nodal reference point sets with a Dirichlet style Boundary condition according to a desired numerical value.
• Kinematically constraining the reference points to the three orthogonal planes, so that the faces deform co-operatively with the deforming reference points.

4.0 Post-test analysis considerations

Finally, I had to think of a good way to undertake the post-test analysis. I asked the solver (ABAQUS) to output reaction forces and displacements from the three reference points. These were then subsequently used to determine stress and strain measures. Stress was determined as the division of the reaction force and the corresponding surface area. Due to the irregular nature of the surface upon which the area was determined, I had to determine this area using a Python script designed to extract this information directly from ABAQUS.

Conclusions

The above represent a set of some challenges you must address if you are to benefit from the Triaxiality challenge. I know this might be a little bit too indepth for some of my readers, but if you find them helpful let me know. I know I might be speaking only to a small subset of this audience but for their sake, let us indulge them with such a dense explanation.

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