ellipsoidal fiber distribution¶

Module: solid

Category: material

Type string: "ellipsoidal fiber distribution"

Parameters¶

Name	Description	Default	Units
`density`	density	1	[M/L^3]
`beta`	beta		[]
`ksi`	ksi		[P]
`mat_axis`			[]

Description¶

The material type for an ellipsoidal continuous fiber distribution is ellipsoidal fiber distribution. Since fibers can only sustain tension, this material is not stable on its own. It must be combined with a stable compressible material that acts as a ground matrix, using a solid mixture container as described in solid mixture.

The Cauchy stress for this fibrous material is given by ¹²³:

\[ \boldsymbol{\sigma}=\int_{0}^{2\pi}\int_{0}^{\pi}H\left(I_{n}-1\right)\sigma_{n}\left(\mathbf{n}\right)\sin\varphi\,d\varphi\,d\theta. \]

Here, $I_{n}=\lambda_{n}^{2}=\mathbf{N}\cdot\mathbf{C}\cdot\mathbf{N}$ is the square of the fiber stretch $\lambda_{n}$, $\mathbf{N}$ is the unit vector along the fiber direction, in the reference configuration, which in spherical angles is directed along $\left(\theta,\varphi\right)$, $\mathbf{n}=\mathbf{F}\cdot\mathbf{N}/\lambda_{n}$, and $H\left(.\right)$ is the unit step function that enforces the tension-only contribution.

The fiber stress is determined from a fiber strain energy function,

\[ \sigma_{n}=\frac{2I_{n}}{J}\frac{\partial\Psi}{\partial I_{n}}\mathbf{n}\otimes\mathbf{n}, \]

where in this material,

\[ $\Psi\left(\mathbf{n},I_{n}\right)=\xi\left(\mathbf{n}\right)\left(I_{n}-1\right)^{\beta\left(\mathbf{n}\right)}. \]

The material parameters $\beta$ and $\xi$ are assumed to vary ellipsoidally with $\mathbf{n}$, according to

\[ \begin{aligned}\xi\left(\mathbf{n}\right) & =\left(\frac{\cos^{2}\theta\sin^{2}\varphi}{\xi_{1}^{2}}+\frac{\sin^{2}\theta\sin^{2}\varphi}{\xi_{2}^{2}}+\frac{\cos^{2}\varphi}{\xi_{3}^{2}}\right)^{-1/2}\\ \beta\left(\mathbf{n}\right) & =\left(\frac{\cos^{2}\theta\sin^{2}\varphi}{\beta_{1}^{2}}+\frac{\sin^{2}\theta\sin^{2}\varphi}{\beta_{2}^{2}}+\frac{\cos^{2}\varphi}{\beta_{3}^{2}}\right)^{-1/2} \end{aligned}\,. \]

Example:

<material id="1" type="solid mixture">
  <mat_axis type="local">0,0,0</mat_axis>
  <solid type="neo-Hookean">
    <E>1000.0</E>
    <v>0.45</v>
  </solid>
  <solid type="ellipsoidal fiber distribution">
    <ksi>10, 12, 15</ksi>
    <beta>2.5, 3, 3</beta>
  </solid>
</material>

Lanir, Y., "Constitutive equations for fibrous connective tissues", J Biomech 16, 1 (1983), pp. 1-12. ↩
Ateshian, G. A., "Anisotropy of fibrous tissues in relation to the distribution of tensed and buckled fibers", J Biomech Eng 129, 2 (2007), pp. 240-9. ↩
Ateshian, G. A., Rajan, V., Chahine, N. O., Canal, C. E., and Hung, C. T., "Modeling the matrix of articular cartilage using a continuous fiber angular distribution predicts many observed phenomena", J Biomech Eng 131, 6 (2009), pp. 061003. ↩