fiber-exp-linear

Module: solid

Category: material

Type string: "fiber-exp-linear"

Parameters

Name	Description	Default	Units
density	density	1	[M/L^3]
c3	c3	0	[P]
c4	c4	0	[]
c5	c5	0	[P]
lambda	lambda	1	[]
fiber			[]

Description

This material type is fiber-exp-linear. This constitutive fiber model has an initial exponential rise and then grows linearly after a user-specified stretch transition point. This fiber material is based on the trans-iso Mooney-Rivlin model introduced in ¹. This material by itself is not stable, so it is recommend to use it as part of a solid mixture.

The strain energy is as follows:

\[ F_{2}\left(\lambda\right)=\begin{cases} 0 & \lambda<1\\ C_{3}\left(e^{-C_{4}}\left(\text{Ei}\left(C_{4}\lambda\right)-\text{Ei}\left(C_{4}\right)\right)-\ln\lambda\right) & 1\le\lambda<\lambda_{m}\\ C_{5}\left(\lambda-\lambda_{m}\right)+C_{6}\ln\frac{\lambda}{\lambda_{m}}+C_{3}\left(e^{-C_{4}}\text{Ei}\left(C_{4}\lambda_{m}\right)-\text{Ei}\left(C_{4}\right)-\ln\lambda_{m}\right) & \lambda_{m}\le\lambda \end{cases}\, \]

where \(\text{Ei}\left(\cdot\right)\) is the exponential integral function. The resulting fiber stress is evaluated from

\[ \lambda\frac{\partial F_{2}}{\partial\lambda}=\begin{cases} 0 & \tilde{\lambda}\leqslant1\\ C_{3}\left(e^{C_{4}\left(\lambda-1\right)}-1\right) & 1<\tilde{\lambda}<\lambda_{m}\\ C_{5}\lambda+C_{6} & \tilde{\lambda}\geqslant\lambda_{m} \end{cases}\,. \]

Here, \(\lambda_{m}\) is the stretch at which the fibers are straightened, \(C_{3}\) scales the exponential stresses, \(C_{4}\) is the rate of uncrimping of the fibers, and \(C_{5}\) is the modulus of the straightened fibers. \(C_{6}\) is determined from the requirement that the stress is continuous at \(\lambda_{m}\) (see below).

While this material enforces continuity of the strain energy density and stress at \(\lambda_{m}\), it does not guarantee continuity of the elasticity. The continuity of the elasticity is only satisfied if the parameter \(C_{3}\) satisfies

\[ C_{3}=\frac{C_{5}}{C_{4}}e^{-C_{4}\left(\lambda_{m}-1\right)} \]

The parameter \(C_{6}\) is chosen so that the stress defined above is continuous \(\lambda_{m}\) and is determined by,

\[ C_{6}=C_{3}\left(e^{C_{4}\left(\lambda_{m}-1\right)}-1\right)-C_{5}\lambda_{m} \]

To use the form of \(F_{2}\) that satisfies continuity of the elasticity at \(\lambda_{m}\) the user may set \(C_{3}=0\) in the input file, and specify \(C_{4}\) and \(C_{5}\) as explained above. When \(C_{3}=0\) the code will automatically recalculate \(C_{3}\) internally using the formula above.

Example:

<material id="1" name="Material" type="solid mixture">
  <solid type="neo-Hookean">
    <E>1e-6</E>
    <v>0</v>
  </solid>
  <solid type="fiber-exp-linear">
    <c3>0.0023</c3>
    <c4>43</c4>
    <c5>3</c5>
    <lambda>1.05</lambda>
    <fiber type="vector">1,0,0</fiber>
  </solid>
</material>

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1. Weiss, J.A., Maker, B.N., and Govindjee, S., "Finite element implementation of incompressible, transversely isotropic hyperelasticity", Computer Methods in Applications of Mechanics and Engineering 135 (1996), pp. 107-128 ↩