Viscoelastic Material

Constitutive Laws

This MPM material has separate constitutive laws for deviatoric stress and pressure.

Deviatoric Constitutive Law

The deviatoric constitutive law is always a small-strain, linear viscoelastic material with time-dependent shear modulus, G(t), which is given 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]

The updates for components of the deviatoriic stress become

      [math]\displaystyle{ ds_{ij} = 2\left( G(0) de_{ij} - \sum_{k=1}^n G_k d\alpha_{ij,k} \right) }[/math]

where α_ij,k are a series of internal variables that are tracked in history variables on each particle.

Pressure Constitutive Law

The pressure constitutive law has two options. The first in to use a small strain linear elastic law with pressure increment of

      [math]\displaystyle{ dP = -K(d\varepsilon_{xx} + d\varepsilon_{yy} + d\varepsilon_{zz} - 3d\varepsilon_{res}) }[/math]

where K is the time-independent bulk modulus and other terms are applied or residual strain increments. The use this law, which is the default, set pressureLaw to 0 and enter the bulk modulus K.

The second option is to use then Mie-Grüneisen equation of state (MGEOS). The use this law, which is only allowed in OSParticulas, set pressureLaw to 1 and enter the MGEOS parameters C0, gamma0, S0, S1, S2, and S3.

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
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.	time units	none
alpha	Thermal expansion coefficient	ppm/K	40
pressureLaw	Picks the constitutive law use for time independent pressure. The options are 0 to linear elastic law and 1 to use MGEOS equation of state.	none	0
K	Time-independent bulk modulus (when using linear elastic law)	pressure units	none
C0	The bulk wave speed (when using MGEOS law)	alt velocity units	4004
gamma0	The γ0 parameter (when using MGEOS law)	none	1.64
S0	The S0 parameter (when using MGEOS law)	none	1.35
S1	The S1 parameter (when using MGEOS law)	none	0
S2	The S2 parameter (when using MGEOS law)	none	0
S3	The S3 parameter (when using MGEOS law)	none	0
(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.