The Cauchy stress for this fibrous material is given by [38, 7, 4]: $$\boldsymbol{\sigma}=\int_{0}^{2\pi}\int_{0}^{\pi}H\left(I_{n}-1\right)\sigma_{n}\left(\mathbf{n}\right)\sin\varphi\,d\varphi\,d\theta\,.$$ 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 (Section 4.10↓). The solid-bound molecule must be defined in the <Globals> section (Section 3.4.3↑) 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>