Fiber in a transversely isotropic biphasic material

**ateshian** · 05-17-2017, 07:35 PM

Hi Antoine,

It is acceptable to embed <fiber> or <mat_axis> tags within the individual <solid> components of a mixture. If you also have a global material axis orientation (e.g., defined in the <ElementData> section), FEBio will compound the transformations. In other words, the <fiber> or <mat_axis> orientation in the <solid> tag will be set relative to the global material axis orientation for that element.

Please note that, as I was double-checking this feature in FEBio, I found a bug in the current code which needed to be fixed. We will be releasing a new version of the code shortly, so please use that latest version.

For the material you define in your example, please note that a "spherical fiber distribution" is technically isotropic. In other words it shouldn't matter which way the material axes are pointing for this type of fiber material. If you do see a difference in your analyses, they should be small and due to the error inherent to numerically integrating the stress over all fiber directions.

Best,

Gerard

**Antoine** · 06-09-2017, 05:07 AM

Hi professor Ateshian. Thank you for your kind reply.

Following the above way of implementing fiber directions, I created a knee model with articular cartilage and meniscus composed of anisotropic biphasic materials, and used an ellipsoidal fiber distribution to model their anisotropic properties.

Would you kindly comment on the values that I used for beta and ksi ? I have conducted some sensitivity analyses for different of beta and Ksi on knee joint sontact stress to understand how they afftect our results. I am happy to share these results with you, if you wish. I drew inspiration from your paper to choose their values: Modeling the matrix of articular cartilage using a continuous fiber angular distribution predicts many observed phenomena.
The description of my materials are in the following posts.

Regarding the meniscus, in order to model the fact that the circumferentially-oriented fibres are stiffer than the radially-oriented one, I increased the value of ksi according to the Y-axis direction (In the coordinate system generated by the mat axis function). Is this modification actually representing a stiffer fiber orientation according Y?

The results seem to be relevant, and match the results obtained in Vitro and with the previous model developed with Abaqus. (Picture)

I also ran a model where I inversed the value of ksi (4,2,1, rather than 2,4,1) for the meniscus and the results were quite similar to the one of the first model. Could we conclude that the fiber orientation does not have much impact on the result, or is it only linked with my model?

I work with the 2.6.4 version of Febio, and 2.3 version of Preview.
Thanks a lot and best regards,

Antoine

**Antoine** · 06-09-2017, 05:10 AM

Articular Cartilage

<material id="7" name="FemCart" type="biphasic">
<phi0>0.2</phi0>
<fluid_density>1</fluid_density>
<solid type="solid mixture">
<solid type="spherical fiber distribution">
<alpha>0</alpha>
<beta>2.15</beta>
<ksi>3</ksi>
</solid>
<solid type="neo-Hookean">
<density>1</density>
<E>0.5</E>
<v>0</v>
</solid>
</solid>
<permeability type="perm-const-iso">
<perm>0.001</perm>
</permeability>
</material>
<material id="8" name="TibCartTransitionalZone" type="biphasic">
<phi0>0.2</phi0>
<fluid_density>1</fluid_density>
<solid type="solid mixture">
<solid type="spherical fiber distribution">
<alpha>0</alpha>
<beta>2.15</beta>
<ksi>3</ksi>
</solid>
<solid type="neo-Hookean">
<density>1</density>
<E>0.5</E>
<v>0</v>
</solid>
</solid>
<permeability type="perm-const-iso">
<perm>0.001</perm>
</permeability>
</material>
<material id="9" name="TibCartSuperficialZone" type="biphasic">
<phi0>0.2</phi0>
<fluid_density>1</fluid_density>
<solid type="solid mixture">
<solid type="neo-Hookean">
<density>1</density>
<E>0.5</E>
<v>0</v>
</solid>
<solid type="ellipsoidal fiber distribution">
<beta>2.15,2.15,2.15</beta>
<ksi>4,2,2</ksi>
</solid>
</solid>
<permeability type="perm-const-iso">
<perm>0.001</perm>
</permeability>
</material>
<material id="10" name="TibCartDeepZone" type="biphasic">
<phi0>0.2</phi0>
<fluid_density>1</fluid_density>
<solid type="solid mixture">
<solid type="neo-Hookean">
<density>1</density>
<E>0.5</E>
<v>0</v>
</solid>
<solid type="ellipsoidal fiber distribution">
<beta>2.15,2.15,2.15</beta>
<ksi>2,2,4</ksi>
</solid>
</solid>
<permeability type="perm-const-iso">
<perm>0.001</perm>
</permeability>
</material>

**Antoine** · 06-09-2017, 05:11 AM

Meniscus

<material id="11" name="Men1" type="biphasic">
<phi0>0.2</phi0>
<fluid_density>1</fluid_density>
<solid type="solid mixture">
<solid type="neo-Hookean">
<density>1</density>
<E>0.5</E>
<v>0</v>
</solid>
<solid type="ellipsoidal fiber distribution">
<mat_axis type="vector">
<a>-3,-1,-2</a>
<d>0,-2,-1</d>
</mat_axis>
<beta>2.15,2.15,2.15</beta>
<ksi>2,4,1</ksi>
</solid>
</solid>
<permeability type="perm-const-iso">
<perm>0.001</perm>
</permeability>
</material>

**ateshian** · 06-11-2017, 03:46 PM

Hi Antoine,

Would you kindly comment on the values that I used for beta and ksi ? I have conducted some sensitivity analyses for different of beta and Ksi on knee joint sontact stress to understand how they afftect our results. I am happy to share these results with you, if you wish.

The values appear reasonable to me. Please keep in mind that the best way to obtain these values is to fit the model to experimental data. The studies from my lab where we measured anisotropic properties of cartilage in compression and tension were done years before we started using the ellipsoidal fiber distribution model. So I don't have specific values to recommend for you, beyond the values we estimated in the 2009 paper. If you are trying to reproduce a previously validated Abaqus model, you can also try to fit the stress-strain response generated by Abaqus for the previously validated material properties (e.g., simulations of biaxial tests).

Regarding the meniscus, in order to model the fact that the circumferentially-oriented fibres are stiffer than the radially-oriented one, I increased the value of ksi according to the Y-axis direction (In the coordinate system generated by the mat axis function). Is this modification actually representing a stiffer fiber orientation according Y?

That is correct. Increasing ksi along a particular direction should produce stiffer responses along that direction.

Could we conclude that the fiber orientation does not have much impact on the result, or is it only linked with my model?

It is possible that the anisotropy of cartilage in tension is not sufficiently strong to produce a noticeable difference in the stress distribution. To my knowledge, this issue has not been specifically addressed in any published studies. I ran a simple contact analysis of a rigid spherical indenter (radius = 200 mm) against a cartilage slab (30 mm x 30 mm x 2 mm), using a biphasic model for the cartilage with ellipsoidal fiber distribution (very similar to the properties you use above, with ksi=2,4,1 along X,Y,Z). I found that the contact pressure distribution (in the X-Y plane) is almost circular (very little eccentricity), despite the fact that ksi along Y is twice the value of ksi along X. This would seem to support your observation. I don't know if the same conclusions would be reached for meniscus, since I have never performed studies on meniscus myself and I am not able to recommend any representative values of meniscus anisotropic properties for the ellipsoidal fiber distribution model.

Best,

Gerard

Fiber in a transversely isotropic biphasic material

Fiber in a transversely isotropic biphasic material

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