Viscoelastic Material
Constitutive Law
This MPM material is a small-strain, linear viscoelastic material with time-dependent shear modulus, G(t), and a time-independent bulk modulus, K. The time-dependen shear modulus is give by a sum of n exponentials:
[math]\displaystyle{ G(t) = G_0 + \sum_{i=1}^n G_i e^{-t/\tau_i} }[/math]
Here G0 is the long-time shear modulus and the short-time shear modulus is the sum:
[math]\displaystyle{ G(0) = \sum_{i=0}^n G_i }[/math]
Material Properties
The unusual task for this material is to use multiple Gk and tauk properties (all with the same property name) to enter a material with multiple relaxation times.
| Property | Description | Units | Default |
|---|---|---|---|
| K | Time-independent bulk modulus | pressure units | none |
| ntaus | The number of relaxation times. This property is only needed in XML files and must come before any Gk or tauk properties. In scripted files, the number is automatically determined from the number of relaxation times you provide. | none | none |
| G0 | The long term (or fully-relaxed) shear modulus | pressure units | 0 |
| Gk | The shear modulus for the next relaxation time. Enter multiple Gk properties to have multiple relaxation times. | pressure units | none |
| tauk | The the next relaxation time. Enter multiple tauk properties to have multiple relaxation times. | sec | none |
| alpha | Thermal expansion coefficient | ppm/K | 40 |
| (other) | Properties common to all materials | varies | varies |
The total number of Gk and tauk properies must be equal. In XML files, that total number must match the supplied ntaus property.
This material can emulate a liquid (i.e., similar to a Tait Liquid Material) by setting G0 to zero, using a single relaxation time with tauk short (on time scale of the simulation), and setting the one Gk modulus (in pressure units) to:
[math]\displaystyle{ G_1 = 10^{-9}\left({\eta\over \tau_1}\right) }[/math]
where [math]\displaystyle{ \eta }[/math] is desired viscosity of the liquid in cP and [math]\displaystyle{ \tau_1 }[/math] is the chosen relaxation time in seconds.
History Variables
This material tracks internal history variables (one for each relaxation time and each component of stress) for implementation of linear viscoelastic properties, but currently none of these internal variables are available for archiving.