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. is the energy dissipation inequality.
So the tensor 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 . 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
I would think 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.