Arruda-Boyce

Module: solid

Category: material

Type string: "Arruda-Boyce"

Parameters

Name	Description	Default	Units
density	density	1	[M/L^3]
k	bulk modulus	0	[P]
pressure_model	pressure_model	0	[]
mu	initial modulus	0	[P]
N	links	1	[]

Description

This material describes an incompressible Arruda-Boyce model ¹.

The uncoupled strain energy function for the Arruda-Boyce material is given by:

\[ \Psi=\mu\sum\limits_{i=1}^{5}\frac{C_{i}}{N^{i-1}}\left(\tilde{I}_{1}^{i}-3^{i}\right)+U\left(J\right), \]

where, \(C_{1}=\frac{1}{2}\), \(C_{2}=\frac{1}{20}\), \(C_{3}=\frac{11}{1050}\), \(C_{4}=\frac{19}{7000}\), \(C_{5}=\frac{519}{673750}\) and \(I_{1}\) the first invariant of the right Cauchy-Green tensor. The volumetric strain function \(U\) is defined as follows,

\[ U\left(J\right)=\frac{1}{2}k\left(\ln J\right)^{2}\,. \]

This material model was proposed by Arruda and Boyce ¹ and is based on an eight-chain representation of the macromolecular network structure of polymer chains. The strain energy form represents a truncated Taylor series of the inverse Langevin function, which arises in the statistical treatment of non-Gaussian chains. The parameter \(N\) is related to the locking stretch \(\lambda_{L}\), the stretch at which the chains reach their full extended state, \(by \lambda_{L}=\sqrt{N}\).

Example:

<material id="1" type="Arruda-Boyce">
  <mu>0.09</mu>
  <N>26.5</N>
  <k>100</k>
</material>

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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. ↩↩