spherical fiber distribution sbm

Module: multiphasic

Category: material

Type string: "spherical fiber distribution sbm"

Parameters

Name	Description	Default	Units
density	density	1	[M/L^3]
alpha	alpha	0	[]
beta	beta	-6.27744e+66	[]
ksi0	ksi0	-6.27744e+66	[]
rho0	rho0	-6.27744e+66	[]
gamma	gamma	-6.27744e+66	[]
sbm	sbm	-1	[]
mat_axis			[]

Description

The material type for a spherical (isotropic) continuous fiber distribution with fiber modulus dependent on solid-bound molecule content is spherical fiber distribution sbm. 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, the fiber strain energy density is given by

\[ \Psi=\frac{\xi}{\alpha}\left(\exp\left[\alpha\left(I_{n}-1\right)^{\beta}\right]-1\right)\,, \]

where \(\xi>0\), \(\alpha\geqslant0\), and \(\beta\geqslant2\). The fiber modulus is dependent on the solid-bound molecule referential density \(\rho_{r}^{\sigma}\) according to the power law relation

\[ \xi=\xi_{0}\left(\frac{\rho_{r}^{\sigma}}{\rho_{0}}\right)^{\gamma}\,, \]

where \(\rho_{0}\) is the density at which \(\xi=\xi_{0}\).

This type of material references a solid-bound molecule that belongs to a multiphasic mixture. Therefore this material may only be used as the solid (or a component of the solid) in a multiphasic mixture. The solid-bound molecule must be defined in the Globals section and must be included in the multiphasic mixture using a solid_bound tag. The parameter sbm must refer to the global index of that solid-bound molecule. The value of \(\rho_{r}^{\sigma}\) is specified within the solid_bound tag. If a chemical reaction is defined within that multiphasic mixture that alters the value of \(\rho_{r}^{\sigma}\), lower and upper bounds may be specified for this referential density within the solid_bound tag to prevent \(\xi\) from reducing to zero or achieving excessively elevated values.

Note: In the limit when \(\alpha\to0\), the expression for \(\Psi\) produces a power law,

\[ \lim\limits_{\alpha\to0}\Psi=\xi\left(I_{n}-1\right)^{\beta} \]

Note: When \(\beta>2\), the fiber modulus is zero at the strain origin (\(I_{n}=1\)). Therefore, use \(\beta>2\) when a smooth transition in the stress is desired from compression to tension.

Example:

<solid type="solid mixture">
  <solid type="neo-Hookean">
    <E>1000.0</E>
    <v>0.45</v>
  </solid>
  <solid type="spherical fiber distribution sbm">
    <alpha>0</alpha>
    <beta>2.5</beta>
    <ksi0>10</ksi0>
    <gamma>2</gamma>
    <rho0>1</rho0>
    <sbm>1</sbm>
  </solid>
</solid>

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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. ↩