Progress
An algorithm for enforcing a user-defined fiber stretch has been implemented in FEBio2. The implementation is based on the paper by Weiss et. al [1] and uses an iterative method for enforcing the prescribed in-situ fiber stretch while maintaining stress equilibrium. The current implementation only works with transversely isotropic Mooney-Rivlin materials but can easily be expanded to other constitutive models. The examples below show the current capabilities.
Example 1: A constant in-situ fiber stretch of 50% was enforced on a cubical block. The example shows that a constant fiber strain can be achieved while maintaining stress equilibrium.
ImageLink(insitu_block1.png, width=400)
Example 2: A constant in-situ fiber stretch of 3% is prescribed on a cylindrical geometry with the fibers oriented circumferentially (left panel). The first principal stress is shown in the middle panel. A radial cut is introduced which relieves the stresses and introducing the opening angle.
ImageLink(insitu_cylinder1.png, width=600)
[1] Weiss J.A., Gardiner J.C., Ellis B.J., Lujan T.J., Phatak N.S., Three-dimensional finite element modeling of ligaments: Technical aspects, Medical Engineering & Physics 27 (2005) 845-861
Test problem implementation in FEBio
The implementation of the in-situ stretch was tested with the proposed test problem. The geometry was created in PreView and the analysis was done as a two-step analysis. In the first step, the in-situ strain is enforced using the algorithm described above. In the next step a saw-tooth prescribed displacement was applied on the free end of the beam. The model uses a trans-iso Mooney-Rivlin material and the stress-displacement is shown below for different levels of in-situ stretch.
attachment:test_results.png
-- ["aerdemir"] DateTime(2013-12-20T09:35:05Z) It seems like the material model is not tension only. If it is, I would suspect to see zero force/stress based on in situ strain level. Also, for the in situ strain case of 0.1 (if nominal) zero length of the object will be around 9.09 mm. For -1 mm displacement from reference length (10 mm), I would suspect the force/stress be less then zero as the nominal strain will be approximately -0.01. According to the plot above, the force/stress drops to zero at a displacement of ~ -1.75 mm. Any insight on this will be helpful.
-- ["belgiansteve"] DateTime(2014-02-23T18:52:07Z) I've found a problem with the initial test runs. The in-situ strain was being enforced during the entire analysis instead of during the first step. This fix changed the results significantly. I've also confirmed that for the case 0 in the figure the result is now identical to the regular trans-iso MR material in FEBio (which was not the case either in the previous runs).
Another test problem
A repeat of test 1 but with different material parameters. The parameters are: c1 = 4.36, c2=0, c3=2.4, c4=30.6, c5=323, k=100, lambda=1.055.
attachment:stress-displacement3.png
Three-Dimensional Ligament Problem
Solution of a three-dimensional ligament problem will be reproduced (see [http://www.ncbi.nlm.nih.gov/pubmed/16085446 Weiss et al., 2005]).
-- ["aerdemir"] DateTime(2014-02-26T20:17:27Z) Steve, do you mind summarizing this problem with links to the manuscript in here?
In this problem we match the experimentally measured in-situ stretch for the MCL. The measured in-situ stretches were input as the target stretches for the pre-strain trans-iso Mooney-Rivlin material. Using the augmented Lagrangian method as described above the fiber stretches were determined to match the target stretches. The geometry and material parameters were taken from the original Nike3D model (described in the paper). The target stretches were actually the final stretches from the Nike3D model (since I couldn't find the corresponding Topaz file). The results produce by FEBio matched the Nike3D results excellently. (see figure below, (left) NIKE3D result, (right) FEBio).
attachment:mcl.jpg
Cardiovascular Test Problem for Generalization
A more challenging test problem, relevant to cardiovascular research, can be found in work by [http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2891240/ Cardamone et al. (2009)]. In this study a mixture model of an arterial segment (see [http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2891240/figure/F1/ Figure 1 of Cardamone et al., 2009]), with an axial pre-stretch and a circumferential residual stress, was explored. Generalization of in situ strain feature of FEBio to such problems will likely extend its userbase, who may be interested in cardiovascular simulations. Model parameters are provided in [http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2891240/table/T1/ Table 1 of Cardamone et al. (2009)].