Implementing an ellipsoidal fiber distribution

[Adil7Khan]

I am trying to implement the constitutive model described in Storm et al to describe a collagen-based ECM. (Storm, Cornelis, et al. "Nonlinear elasticity in biological gels." Nature 435.7039 (2005): 191-194) (See equation in attached picture).

The equations for stress and elasticity tensors consider averaging over probability distribution P(r). In my case, I can model P(r) as an ellipsoidal volume matching the 3D distribution of collagen fibers. My question is how do I calculate the average over P(r) while writing a plugin? Is there a way to incorporate the ellipsoidal fiber distribution as described in the User Manual for the uncoupled continuous fiber distribution?

Thanks a lot,
Adil
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[ateshian]

Hi Adil,

I perused the paper and I am not sure what the tensor Lambda represents in that formula. The paper calls is the "local (not necessarily symmetric) Cauchy deformation tensor." I looked up "Cauchy deformation tensor" and Wikipedia also calls it the Piola strain tensor, which is shown on that page to be the inverse of the left Cauchy-Green tensor (which is symmetric). Based on the rest of the definitions in the paper, I would rather assume that their "Cauchy deformation tensor" is what we call the deformation gradient F (which is not symmetric in general).

In that case, the formula of their paper (and the one you shared) is identical to the definition of continuous fiber distributions given in the User's Manual. Note that the User's Manual section provides the evaluation of the strain energy density, not the stress. To calculate the stress in a fiber from the strain energy density, see (for example) this description in the Theory Manual. A careful comparison of their terms with those of the Theory Manual section should give you a better understanding of how they are equivalent.

This means that you can use the existing continuous fiber distribution material to represent the model from that paper. The only caveat might be the exact form of the function f given in the above formula, versus the various fiber materials currently available in FEBio. If you don't find a material that reproduces the model from Storm et al., you can define a new fiber material derived from FEFiberMaterial or FEFiberMaterialUncoupled.

Best,

Gerard

[Adil7Khan]

Thanks Dr. Ateshian,

Yes, you are right. Lambda in the formula does represent deformation gradient F.

As for the functional form of f, I was planning on using fiber with exponential power law (the form used here works perfectly for me). So if I am understanding this correctly, I should be able to use the uncoupled continuous fiber distribution model with the fibers type="fiber-exponential-power-law-uncoupled", right?

My other question is on how to include the parameter rho in the equation from Storm et al (the term rho is the number of links/fibers per unit volume)? Should I just account for it in the parameter ksi of the fiber material?

Thanks a lot,
Adil

[ateshian]

Hi Adil,

Yes, if the parameter rho is a constant you can factor it into ksi.

Best,

Gerard