EFD uncoupled

Module: solid

Category: material

Type string: "EFD uncoupled"

Parameters

Name	Description	Default	Units
density	density	1	[M/L^3]
k	bulk modulus	0	[P]
pressure_model	pressure_model	0	[]
beta	beta		[]
ksi	ksi		[P]
mat_axis			[]

Description

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

The stress \(\tilde{\boldsymbol{\sigma}}\) for this material is given by ¹²³:

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

\(\tilde{I}_{n}=\tilde{\lambda}_{n}^{2}=\mathbf{N}\cdot\mathbf{\tilde{C}}\cdot\mathbf{N}\) is the square of the fiber stretch \(\tilde{\lambda}_{n}\), \(\mathbf{N}\) is the unit vector along the fiber direction (in the reference configuration), which in spherical angles is directed along \(\left(\theta,\varphi\right)\), \(\mathbf{n}=\mathbf{\tilde{F}}\cdot\mathbf{N}/\tilde{\lambda}_{n}\), 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 in the usual manner,

\[ \tilde{\sigma}_{n}=\frac{2\tilde{I}_{n}}{J}\frac{\partial\tilde{\Psi}}{\partial\tilde{I}_{n}}\mathbf{n}\otimes\mathbf{n}\,, \]

where in this material,

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

The materials parameters \(\beta\) and \(\xi\) are determined from:

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

Example:

<material id="1" type="uncoupled solid mixture">
  <mat_axis type="local">0,0,0</mat_axis>
  <k>1000</k>
  <solid type="Mooney-Rivlin">
    <c1>1</c1>
    <c2>0</c2>
  </solid>
  <solid type="EFD uncoupled">
    <ksi>10, 12, 15</ksi>
    <beta>2.5, 3, 3</beta>
  </solid>
</material>

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1. Lanir, Y., "Constitutive equations for fibrous connective tissues", J Biomech 16, 1 (1983), pp. 1-12. ↩

2. 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. ↩

3. 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. ↩