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uncoupled fiber-entropy-chain

Module: solid

Category: material

Type string: "uncoupled fiber-entropy-chain"

Parameters

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

Description

This material type is uncoupled fiber-entropy-chain. It was proposed by 1. 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

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

where \(\tilde{I}_{n}=\mathbf{n}_{r}\cdot\tilde{\mathbf{C}}\cdot\mathbf{n}_{r}\) is the deviatoric measure 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="uncoupled solid mixture">
    <k>1e4</k>
    <solid type="Mooney-Rivlin">
      <c1>10.0</c1>
      <c2>0</c2>
    </solid>
    <solid type="uncoupled fiber-entropy-chain">
      <N>2.3</N>
      <ksi>1</ksi>
      <n_term>25</n_term>
      <fiber type="vector">0,0,1</fiber>
    </solid>
</material>

  1. Shi, Lei, Hu, Lingfeng, Lee, Nicole, Fang, Shuyang, and Myers, Kristin, "Three-dimensional anisotropic hyperelastic constitutive model describing the mechanical response of human and mouse cervix", Acta Biomaterialia 150 (2022), pp. 277--294. ↩