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I work with PEG hydrogels and use the material to recapitulate cartilage biology and I am really interested in modeling the soft tissue mechanics in COMSOL. I have been studying the work of Caccavo et al closely as the work implements both poroelastic and viscoelastic physics (https://doi.org/10.1016/j.msec.2017.02.155). I understand most of the derivations in the manuscript but I am hung up one caveat.

The paper uses momentum and mass balances along with a volumetric constraint equation as the basis for the model. The energy inequality is used to close the system. The paper poses the problem as axisymmetric to reduce computational complexity and here is where I go off the rails. The energy inequality generates the following kinematic equation for the dashpot which is used to model the viscoelastic behavior of the material. A is the energy dissipation inequality.

F¯¯visct=1ηAF¯¯visc

So the tensor F¯¯visc is describing the impact of the dashpot. In a 2d problem (r,z) I would expect this tensor to be a 2x2. I would not need to give any consideration to the theta direction due to symmetry. Interestingly that is not what the paper does. In the implementation there are 5 equations needed for the kinematics. I took this to be each equation to describe an element of the tensor F¯¯visc. The author has a YouTube video that has been helpful in showing how this model can be entered into COMSOL and solved. He sets up these five equations at the 1:50 mark. He sets up the following with E=F¯¯visc

[E110E130E220E310E33]

I would think E22 should be zero and thus I need only 4 equations for the dashpot kinematics.

I would be grateful to anyone who could clear up my misconception of how this works.

Thanks in advance.

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1 Answer 1

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Imagine a cylinder, with the axis of symmetry aligned with the y axis.

I could rotate the system around either the x or z axes. For these kinds of rotations, I need to consider the moment of inertia to be a tensor, since there is not a symmetry for rotations around these axes.

But, I could also rotate the system around the y axis. This is a very simple kind of rotation, since the object is symmetric for rotations around y, and therefore the moment of inertia for rotations about this axis is just a number, which is analogous to your E22. If I set this number to zero, aka set the moment of inertia for rotations about the y axis to zero, I would get non-sensical results, for example an infinite angular acceleration for any torque applied to the cylinder around the y axis.

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