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Arruda-Boyce unconstrained

Module: solid

Category: material

Type string: "Arruda-Boyce unconstrained"

Parameters

Name Description Default Units
density density 1 [M/L^3]
ksi ksi 1e-05 []
N N 100 []
n_term n_term 30 []
kappa kappa 0 []

Description

The material type for an unconstrained Arruda-Boyce material is Arruda-Boyce unconstrained. This isotropic Arruda-Boyce 1 8-chain model has been used in the field of polymer modeling and in cervical biomechanics2. It is based on the simple freely-jointed-chain model and can be derived using the Langevin equation of statistical mechanics. The strain energy density for this material takes the form

\[ \Psi(I_{1},J)=\xi\left(\sqrt{\frac{I_{1}}{3N}}\beta+\ln\frac{\beta}{\sinh\beta}\right)-\frac{\xi}{3}\sqrt{N}\beta_{0}\ln J+\frac{1}{2}\kappa(\ln J^{2}-1-2\ln J) \]

where \(I_{1}=tr\mathbf{C}\) is the first invariant of \(\mathbf{C}\), \({\displaystyle \beta=\mathcal{L}^{-1}\left[\sqrt{\frac{I_{1}}{3N}}\right]}\) is the inverse Langevin equation, and \({\displaystyle \beta_{0}=\mathcal{L}^{-1}\left[\sqrt{\frac{1}{N}}\right]}\). Here, \(\xi=nk\Theta\) is the initial fiber modulus with \(n\) \(k\), \(\Theta\) 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 that represents the extensibility of the material.

A Taylor series expansion is used to evaluate the inverse Langevin equation. The parameter \(n_{\mathrm{term}}\) is used to control the number of terms used to evaluate the series; \(n_{\mathrm{term}}\) must be an integer between 3 and 30.

Langevin mechanics describes a stochastic process with non-Gaussian distribution. It will reduce to a Gaussian distribution when the system approaches equilibrium. Thus, when the material is far from its maximum extensibility, i.e., when \(\zeta=\sqrt{N}\) is much larger than the stretch, this model will reduce to a neo-Hookean material.

Example: 

<material id="1" name="Soft Tissue" type="Arruda-Boyce unconstrained">
    <N>5</N>
    <ksi>1</ksi>
    <n_term>30</n_term>
    <kappa>1</kappa>
</material>

  1. Arruda, E.M. and Boyce, M.C., "A Three-Dimensional Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials", J. Mech. Phys. Solids 41, 2 (1993), pp. 389-412. ↩

  2. 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. ↩