uncoupled viscoelastic¶

Module: solid

Category: material

Type string: "uncoupled viscoelastic"

Parameters¶

Name	Description	Default	Units
`density`	density	1	[M/L^3]
`k`	bulk modulus	0	[P]
`pressure_model`	pressure_model	0	[]
`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¶

These materials produce a viscoelastic response only for the deviatoric stress response. They must be used whenever the elastic response is uncoupled. The material type for these materials is uncoupled viscoelastic.

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

\[ \mathbf{S}\left(t\right)=pJ\,\mathbf{C}^{-1}+J^{-2/3}\int\limits_{-\infty}^{t}G\left(t-s\right)\frac{d\left(\text{Dev}\left[\mathbf{\tilde{S}}^{e}\right]\right)}{ds}\,ds\,, \]

where \(\mathbf{\tilde{S}}^{e}\) is the elastic stress derived from \(\tilde{W}\left(\mathbf{\tilde{C}}\right)\) 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.

Example:

<material id="1" name="Material 1" type="uncoupled viscoelastic">
  <g1>0.95</g1>
  <t1>0.01</t1>
  <k>100</k>
  <elastic type="Mooney-Rivlin">
    <c1>1</c1>
    <c2>0.0</c2>
  </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. ↩