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. To 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). To use this law, set pressureLaw to 1 and enter the MGEOS parameters C0, gamma0, 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
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
S1	The S1 parameter (when using MGEOS law)	none	1.35
S2	The S2 parameter (when using MGEOS law)	none	0
S3	The S3 parameter (when using MGEOS law)	none	0
Kmax	Maximum bulk modulus (relative to zero-pressure bulk modulus) allowed	none	20
G0	The long term (or fully-relaxed) shear modulus	pressure units	0
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
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 (ignored when using MGEOS law)	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.

Viscoelastic Solids and Liquids

If G0 is not zero, the material is a viscoelastic solid, which means the shear stress at infinite time reamains a finite number. Viscoelastic solids are used to model materials such as elastomers that do not show long time flow due to their cross links or have a plateau shear modulus equal to G0.

If G0 is zero, the material is a viscoelastic liquid that will flow like a liquid if you wait long enough. For example, to emulate a liquid (i.e., similar to a Tait Liquid Material), set G0 to zero, use a single relaxation time with tauk short (on time scale of the simulation), and set the one Gk modulus to:

      [math]\displaystyle{ G_1 = {\eta\over 2\tau_1} }[/math]

where [math]\displaystyle{ \eta }[/math] is desired viscosity and [math]\displaystyle{ \tau_1 }[/math] is the single relaxation time.

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.

This material also tracks J (total relative volume change) and J_res (volume change of free expansion state) as history variables 1 and 2. Note that J_res is only needed, and therefore only tracked, when using MGEOS for pressure constitutive law (when pressureLaw is 1). If not tracked, it is always 1.