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 deviatoric 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 αij,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 properties.
Note that this material always models bulk modulus as a time-independent property (although possible nonlinear elastic). You can model isotropic materials that have a time-dependent bulk modulus by treating them as a special case of a transversely isotropic material.
Plane Stress Analysis
This material can be used in plane stress analysis, but only if it uses the linear pressure law (pressureLaw=0) and it does not add artificial viscosity. Support for plane stress in other conditions may be provided soon.
Effective Time Implementation
The above constitutive law assumes G(t) depends only on time, but in real materials, it will depend strongly on temperature. When modeling diffusion with solvent concentration, it may depend on solvent concentration as well. These dependencies are modeling by assuming the material obeys time-temperature or time-solvent superposition whereby G(t) is given by:
[math]\displaystyle{ G(t) = G_0 + \sum_{i=1}^n G_i e^{-t/(a\tau_i)} }[/math]
where a is a shift factor. In other words, the coefficient of G(t) remain constant, but the relaxation times shift by a factor a. We are lead to define relaxation using a reduced time, which is effective time at some reference condition, as
[math]\displaystyle{ G(t_r) = G_0 + \sum_{i=1}^n G_i e^{-t/\tau_i} \quad{\rm and}\quad G(t) = G(t_r/a) }[/math]
The above constitutive law can then be extended to variable environment by converting convolution integrals to integrals of effective time defined by
[math]\displaystyle{ t_r = \int_0^t \frac{dt}{a} \quad{\rm and}\quad \frac{dt_r}{dt} = \frac{1}{a} }[/math]
To to implement this modeling, the material methods need input for a as a function of environment. When both temperature and solvent vary, a is replaced by aTam'. These shift factors are explained in the next two sections.
Variable Temperature
For temperature variations, aT is the thermal shift factor. In polymer materials, this shift factor is approximated by the WLF equation:
[math]\displaystyle{ \log a_T = \log \frac{\tau(T)}{\tau(T_{ref})} = - \frac{C_1(T-T_{ref})}{T-T_{ref}+C_2} }[/math]
This shift factor implemented in simulations by entering Tref, C1, and C2. In polymer materials, if Tref=Tg, or the glass transition temperature, C1=17.44 and C2=51.6 are average values of a range of polymers (the values for one specific polymer, however, may vary significantly). If Tref differs from Tg, it can be used in a shifted WLF equation using C3 and C4 defined from C1 and C2 at Tg by
[math]\displaystyle{ \log \frac{\tau(T)}{\tau(T_{ref})} = \log a_T = - \frac{C_3(T-T_{ref})}{T-T_{ref}+C_4} }[/math]
where
[math]\displaystyle{ C_3 = \frac{C_1C_2}{C_2+T_{ref}-T_g} \quad{\rm and}\quad C_4 = C_2+T_{ref}-T_g }[/math]
Variable Solvent Concentration
For concentration variations, am is the solvent shift factor. In absence of a WLF equation, the solvent shift is modeled using a WLF-style equation based on assumption that solvent expansion leads to free volume that promotes relaxation:
[math]\displaystyle{ \log a_m = \log \frac{\tau(c)}{\tau(c_{ref})} = - \frac{C_{m1}(c-c_{ref})}{c+C_{m2}} }[/math]
This shift factor implemented in simulations by entering mref, Cm1, and Cm2. Note that unlike the WLF equation, this equation does not include [math]\displaystyle{ c_{ref} }[/math] in the denominator. Based on free-value arguments, a reasonable estimate for [math]\displaystyle{ C_{m2} }[/math] is:
[math]\displaystyle{ C_{m2} = \frac{0.025}{\beta_V} }[/math]
where [math]\displaystyle{ \beta_V }[/math] is the volumetric solvent expansion coefficient. If [math]\displaystyle{ C_{m1} }[/math] is known at one [math]\displaystyle{ c_{ref} }[/math], the value at a new reference concentration, [math]\displaystyle{ c_{ref}' }[/math] would be:
[math]\displaystyle{ C_{m1}' = \frac{C_{m1}C_{m2}}{C_{m2}+c_{ref}'-c_{ref}} }[/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 |
|---|---|---|---|
| pressureLaw | Picks the constitutive law used 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 |
| (MGEOS) | Enter MGEOS properties C0, S1, S2, S3, gamma, and Kmax. The UJOption is fixed at 1. | varies | varies |
| 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 next relaxation time. Enter multiple tauk properties to have multiple relaxation times. | time units | none |
| Tref | Reference temperature to shift relaxation times using the WLF equation. If Tref<0, the not shifting is done and relaxation times will be independent of temperature. | K | -1 |
| C1 | Coefficient in WLF equation used when Tref≥0 to shift relaxation times | none | 17.44 |
| C2 | Coefficient in WLF equation used when Tref≥0 to shift relaxation times | none | 51.6 |
| 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.
The default value for Kmax is -1, which means to not limit the bulk modulus. This mode is almost always stable, but simulations with high compression should always add the AdjustTimeStep Custom Task to keep calculation stable under high tangent bulk modulus conditions.
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 Jres (volume change of free expansion state) as history variables 1 and 2. Note that Jres 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.