viscoelastic¶

Module: solid

Category: material

Type string: "viscoelastic"

Parameters¶

Name	Description	Default	Units
`density`	density	1	[M/L^3]
`t1`	t1	1	[t]
`t2`	t2	1	[t]
`t3`	t3	1	[t]
`t4`	t4	1	[t]
`t5`	t5	1	[t]
`t6`	t6	1	[t]
`g0`	g0	1	[]
`g1`	g1	0	[]
`g2`	g2	0	[]
`g3`	g3	0	[]
`g4`	g4	0	[]
`g5`	g5	0	[]
`g6`	g6	0	[]
`elastic`			[]

Description¶

The material type for viscoelastic materials is viscoelastic.

For a viscoelastic material, the second Piola Kirchhoff stress can be written as follows ¹:

\[ \mathbf{S}\left(t\right)=\int\limits_{-\infty}^{t}G\left(t-s\right)\frac{d\mathbf{S}^{e}}{ds}\,ds\,, \]

where \(\mathbf{S}^{e}\) is the elastic stress and \(G\) is the relaxation function. It is assumed that the relaxation function is given by the following discrete relaxation spectrum:

\[ G\left(t\right)=\gamma_{0}+\sum\limits_{i=1}^{N}\gamma_{i}\exp\left(-t/\tau_{i}\right)\,, \]

Note that the user does not have to enter all the \(\tau_{i}\) and \(\gamma_{i}\) coefficients. Instead, only the values that are used need to be entered. So, if \(N\) is 2, only \(\tau_{1}\), \(\tau_{2}\), \(\gamma_{1}\) and \(\gamma_{2}\) have to be entered.

The elastic parameter defines the elastic material and must be an unconstrained elastic material.

Example:

<material id="1" name="Material 1" type="viscoelastic">
  <g1>0.95</g1>
  <t1>0.01</t1>
  <elastic type="neo-Hookean">
    <E>1</E>
    <v>0.0</v>
  </elastic>
</material>

Puso, M. A. and Weiss, J. A., "Finite element implementation of anisotropic quasi-linear viscoelasticity using a discrete spectrum approximation", J Biomech Eng 120, 1 (1998), pp. 62-70. ↩