fiber-entropy-chain

Module: solid

Category: material

Type string: "fiber-entropy-chain"

Parameters

Name	Description	Default	Units
density	density	1	[M/L^3]
N	N	-6.27744e+66	[]
ksi	ksi	0	[P]
mu	mu	0	[P]
n_term	n_term	30	[]
fiber			[]

Description

This material type is fiber-entropy-chain. It was proposed by [#Shi22]. This fiber model is based on statistical mechanics to reflect the entropy-driven nature of a biological fiber. The model is derived from the freely-jointed-chain mechanism. Its strain energy is directly related to the entropic change of the chains in the material, given by

\[ \Psi_{n}\left(I_{n}\right)=\xi N\left(\sqrt{\frac{I_{n}}{N}}\beta+\ln\frac{\beta}{\sinh\beta}\right)-\frac{\xi\sqrt{N}}{2}\beta_{0}I_{n}-\alpha_{00} \]

where \(I_{n}=\mathbf{n}_{r}\cdot\mathbf{C}\cdot\mathbf{n}_{r}\) is the square of the stretch ratio along the referential fiber direction \(\mathbf{n}_{r}\). The inverse Langevin equation relates the parameter \(\beta\) to \(I_{n}\) and N according to \({\displaystyle \beta=\mathcal{L}^{-1}\left[\sqrt{\frac{I_{n}}{N}}\right]}\). Here, \(\beta_{0}\) is the value of \(\beta\) when \(I_{n}=1\), and \(\alpha_{00}\) is needed to produce a state of zero energy at \(I_{n}=1\). The parameter \(\xi=nk\Theta\) is the initial fiber stiffness, with \(n\), \(k\), and \(\Theta\) respectively representing the number of chains per unit volume, Boltzmann's constant, and the absolute temperature. The parameter \(N=\zeta^{2}\) is the number of chain segments, and \(\zeta\) is the locking stretch, representing the extensibility of the fiber. The Langevin function is given by \(\mathcal{L}\left(x\right)=\coth x-\frac{1}{x}\).

When evaluating the inverse Langevin equation, a Taylor series expansion is applied for computational stability. The parameter \(n_{\mathrm{term}}\) is used to control the number of terms used to evaluate the equation; it must be an integer between 3 and 30.

Example:

<material id="1" name="Soft Tissue" type="solid mixture">
    <solid type="Arruda-Boyce unconstrained">
      <N>2</N>
      <ksi>1</ksi>
      <n_term>30</n_term>
      <kappa>1</kappa>
    </solid>
    <solid type="continuous fiber distribution">
        <mat_axis type="angles">
            <theta>0</theta>
            <phi>0</phi>
        </mat_axis>
        <fibers type="fiber-entropy-chain">
            <ksi>1</ksi>
            <N>2</N>
            <n_term>30</n_term>
        </fibers>
        <distribution type="von-Mises-3d">
            <b>0.3</b>
        </distribution>
        <scheme type="fibers-3d-fei">
            <resolution>196</resolution>
        </scheme>
    </solid>
</material>