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	MPa	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	MPa	0
Gk	The shear modulus for the next relaxation time. Enter multiple Gk properties to have multiple relaxation times.	MPa	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 MPa) 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.