uncoupled fiber-exp-linear

Module: solid

Category: material

Type string: "uncoupled fiber-exp-linear"

Parameters

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

Description

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:

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

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

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

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 enforce continuity of the elasticity. To enforce continuity of all three parameters, we need to let

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

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 enforce continuity of the elasticity at \(\lambda_{m}\) the user may also 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 above formula to calculate \(C_{3}\) manually for use in the first form can also enforce continuity of the elastic modulus at \(\lambda_{m}\).

Example:

<material id="1" name="Material" type="uncoupled solid mixture">
  <k>10</k>
  <solid type="Mooney-Rivlin">
    <c1>2.5e-07</c1>
    <c2>0</c2>
  </solid>
  <solid type="uncoupled fiber-exp-linear">
    <c3>0.00234</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. ↩