When I got interested in biomechanics, i read the paper "Quasi-linear viscoelastic behavior of the human periodontal ligament" (PDL) of Toms et all 2002 (doi). Thought this is a handy example to get familiar with parameter studies and python scripting in ABAQUS. My objective was a poro-visco-hyperelastic model of the PDL, that reproduces the measured force-deflection curve of the eight specimens. Because I exploited symmetry, model size is fairly small and you can run it with the ABAQUS Learning Edition.
To make a long story short, Toms cut thin sections of the PDL, loaded the eight specimens in transverse direction, measured the force-deflection curve, and did then curve fits using Fung's quasi-linear viscoelastic (QLV) model. With the data in the paper you can reconstruct the fitted reaction force versus displacement curve, which I took as target for my simulation. Toms approximated the specimens as cylindrical ring, which I modeled using axisymmetric elements and symmetry boundary conditions. I coded the dimensions, actuator displacement time history and parameters of the relaxation function (given in the paper) in the ABAQUS input file (.inp). A run of all eight test is then executed using the ABAQUS python script in the parametric study file (.psf). The plot below shows a animation of the pore pressure and pore fluid velocity of specimen A1 for the first 0.13 seconds.
The first step was a visco-hyperelastic simulation, with *MARLOW hyperelastic constitutive model. For the *MARLOW model the uniaxial stress-strain curve is needed. Assume that there is a homogeneous mapping between fitted force-deflection and stress-strain curve. Then, we want to find the parameters of the mapping, that minimizes the L2-Norm of the difference between fitted curve and simulated response (with the actuator ramp input from zero to maximum displacement). For that I used a for loop on a scale factor (call a matlab script to do the mapping and some post processing ...) in the python psf file and evaluated the results for the minimum. For each specimen the optimal stress-strain curve was identified. The plot below shows the stress-relaxation test of the first specimen A1, that lasts 50 minutes (left the middle section out).
The final step is a poro-visco-hyperelastic simulation. Parameters for poroelasticity are from the paper of Bergomi et al. 2011 (doi). Free draining was specified at the top and bottom free cut surface, and towards the alveolar bone at the outer radius. The animation at the top shows the initial response. When I saw that the first time I was puzzled, pore fluid velocities vanish quickly after the ramp input remains constant and pore pressure is negligible. I double checked the usual things (units, ...) and that looked o.k. So I was left with the question, "Is it a bug or is it a feature". With some distance and during a jogging lap the answer appeared, "It is a feature". Why?
Addendum: I have contributed the answer to the discussion in my recent paper at Springer Nature, Scientific Reports, Towards a reduced order model of the periodontal ligament (doi), 'the specimen is predominantly loaded in shear, and it should be noted that pure shear deformation does not involve dilation'. This means that no pressure gradient is built up to drive a Darcy flow.
PS: Find the final version of the input files in (zip). If you want to simulate the 50 minutes relaxation test you better omit the poro part ;-)
