This example is the last in a series illustrating the "cell growth" material in FEBio. To better understand the material presented here, please refer to the examples on Single Cell Growth, Internal Constraints to Growth, and External Constraints to Growth.
This last example demonstrates the interesting combination of growth and buckling. This example is inspired by a presentation I saw by Oliver Jensen, of the University of Nottingham, titled "Multiscale models for tissue growth: from epithelium to epidermis" and given at the Workshop in Microscale Modeling in Biomechanics and Mechanobiology, held at Ericeira, Portugal, May 30-June 1, 2011.
Consider a beam fixed at both ends, consisting of a "solid mixture" with a "neo-Hookean" solid matrix (Young's modulus = 1 kPa) and "cell growth" material, where cells grow fivefold.
(click on image to see movie)
BeamNoBuckling.feb
If there are no other constraints on the beam, the growth will simply consist of an apparent swelling, albeit constrained at the fixed ends.
(click on image to see movie)
BeamBuckling.feb
Now consider a disturbance in the form of a small force, temporarily nudging the beam upward at its center, while growth is proceeding. As a result of this small disturbance, the beam buckles during growth and takes on a very different shape, as shown in the attached movie. This form of growth is reminiscent of the process of invagination during morphogenesis.
(click on image to see movie)
BeamBucklingStiffer.feb
Buckling is an instability which may exhibit multiple modes. In this last example, the Young's modulus of the beam is increased to 10 kPa. All else remaining the same, the resulting buckling behavior exhibits multiple folds, as shown in the attached movie.
This last series of example further demonstrates the rich set of outcomes that may be achieved with the "cell growth" model implemented in FEBio. Buckling analyses under finite deformation are not trivial problems in a finite element framework. The files attached with these examples will exhibit slow convergence, but they should run to completion.
If you try the cell growth model and get cool results, feel free to share them on this forum.
Gerard
This last example demonstrates the interesting combination of growth and buckling. This example is inspired by a presentation I saw by Oliver Jensen, of the University of Nottingham, titled "Multiscale models for tissue growth: from epithelium to epidermis" and given at the Workshop in Microscale Modeling in Biomechanics and Mechanobiology, held at Ericeira, Portugal, May 30-June 1, 2011.
Consider a beam fixed at both ends, consisting of a "solid mixture" with a "neo-Hookean" solid matrix (Young's modulus = 1 kPa) and "cell growth" material, where cells grow fivefold.
(click on image to see movie)
BeamNoBuckling.feb
If there are no other constraints on the beam, the growth will simply consist of an apparent swelling, albeit constrained at the fixed ends.
(click on image to see movie)
BeamBuckling.feb
Now consider a disturbance in the form of a small force, temporarily nudging the beam upward at its center, while growth is proceeding. As a result of this small disturbance, the beam buckles during growth and takes on a very different shape, as shown in the attached movie. This form of growth is reminiscent of the process of invagination during morphogenesis.
(click on image to see movie)
BeamBucklingStiffer.feb
Buckling is an instability which may exhibit multiple modes. In this last example, the Young's modulus of the beam is increased to 10 kPa. All else remaining the same, the resulting buckling behavior exhibits multiple folds, as shown in the attached movie.
This last series of example further demonstrates the rich set of outcomes that may be achieved with the "cell growth" model implemented in FEBio. Buckling analyses under finite deformation are not trivial problems in a finite element framework. The files attached with these examples will exhibit slow convergence, but they should run to completion.
If you try the cell growth model and get cool results, feel free to share them on this forum.
Gerard
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