Tait Liquid Material

Constitutive Law

This MPM material models a liquid as a hyperelastic material. The pressure in the liquid is found from the Tait equation:

      [math]\displaystyle{ V(p,T) = V(0,T)\left[1 - C \ln\left(1+{p\over B(T)}\right)\right] }[/math]

where C = 0.0894 is a universal Tait constant, V(0,T) is the temperature dependence of the volume at zero pressure, and

      [math]\displaystyle{ B(T) = CK(0,T) }[/math]

where K(0,T) is the temperature dependence of the bulk modulus at zero pressure. Defining J as relative volume (i.e., determinant of total deformation gradient) and J_res as determinant of deformation gradient due to free thermal expansion, or:

      [math]\displaystyle{ J = {V(p,T)\over V(0,T_0)} \qquad {\rm and} \qquad J_{res} = {V(0,T)\over V(0,T_0)} = e^{\beta_0(T-T_0)} }[/math]

where T₀ is the stress free temperature and β₀ is the zero-pressure, volumetric, thermal expansion coefficient (which has been assumed to be independent of temperature), the constitutive law for pressure is:

      [math]\displaystyle{ p = CK_0\left[\exp\left({1\over C}\left(1 - {J\over J_{res}}\right)\right)-1\right] }[/math]

Here the zero-pressure bulk modulus is K₀, and it has also been assumed to be independent of temperature (i.e., B(T) = CK₀). This material is equivalent to a hyperelastic material with volumetric strain energy function of

      [math]\displaystyle{ U(J^*) = C K_0\left[ C \exp\left({1-J^*\over C}\right) + J^*\right] }[/math]

where J^* = J/J_res is the effective volumetric ratio. This energy function equals the energy per unit initial volume for isothermal compression or expansion of a Tait liquid.

Shear Stress

For shear stress calculations, this material is assumed to be a Newtonian fluid, which means that the shear stress is proportional to deviatoric, symmetrized velocity gradient:

      [math]\displaystyle{ \tau = \eta(\dot\gamma) \left(\nabla \mathbf{v} + \nabla \mathbf{v}^T - {2\over 3}{\rm Tr}(\nabla \mathbf{v})\mathbf{I}\right) = 2 \eta(\dot\gamma) {\rm dev}(\nabla v) }[/math]

where [math]\displaystyle{ \nabla \mathbf{v} }[/math] is the velocity gradient and [math]\displaystyle{ \eta(\dot\gamma) }[/math] is the viscosity at a given shear rate. The total stress is given by [math]\displaystyle{ \mathbf{\sigma} = -p \mathbf{I} + \tau }[/math].

If the material is assigned only a viscosity, then [math]\displaystyle{ \eta(\dot\gamma) }[/math] in independent of shear rate describing a Newtonian fluid. Alternatively, you can enter any arbitrary shear-rate dependence of viscosity for the material. The shear rate is defined by

      [math]\displaystyle{ \dot\gamma = 2 |{\rm dev}(\nabla v)| }[/math]

Pressure-Dependent Properties

For a Tait liquid, the pressure- and temperature-dependent bulk modulus is

      [math]\displaystyle{ K(P,T) = {p + B(T)\over C} {J\over J_{res}} }[/math]

The pressure- and temperature-dependent, volumetric thermal expansion coefficient is

      [math]\displaystyle{ \beta(P,T) = \beta(0,T) + {P\over K(P,T)B(T)} {dB(T)\over dT} }[/math]

where

      [math]\displaystyle{ \beta(0,T) = {1\over V(0,T)} {dV(0,T)\over dT} }[/math]

is the low-pressure thermal expansion coefficient at temperature T. These are general Tait equation results. When the low-pressure bulk modulus and thermal expansion coefficients are independent of temperature, they simplify to:

      [math]\displaystyle{ K(P,T) = {p + CK_0\over C} {J\over J_{res}} \qquad {\rm and} \qquad \beta(P,T) = \beta_0 }[/math]

Material Properties

The properties for a Tait liquid are:

Property	Description	Units	Default
K	Zero-pressure, bulk modulus	pressure units	none
viscosity	Liquid viscosity at one shear rate	viscosity units	none
logshearrate	log of shear rate corresponding to a viscosity	1/(time units	none
alpha	Linear thermal expansion coefficient (β0 = 3α)	ppm/K	0
InitialPressure	You can set initial pressure to a user defined function of position that evaluates to a pressure in pressure units. When liquids are modeled in gravity, the function can set to ρgh, where h is height of liquid above the position (x, y, z).	pressure units	none
(other)	Properties common to all materials	varies	varies

If one viscosity is given and no logshearrate, the material has constant viscosity. If multiple logshearrate and viscosity commands are used, each pair defines points for piecewise linear representation of viscosity as a function of log shear rate. You must enter an equal number of logshearrate and viscosity commands with monotonically increasing shear rates. The liquid's viscosity will be interpolated within the provided points. Shear rates below the minimum or above the maximum provided shear rates will be equal to the viscosity at the minimum or maximum shear rate, respectively.

Note that when initial pressure is set, the mass of each particle will be adjusted to fill the initial particle volume under the set pressure. The particle deformation tracked during the simulation will be the deformation relative to the initial state under pressure. The particle's relative volume, which is tracked in history variable 1, will be relative to the pressure-free state and thus will include volume change due to the initial pressure.

History Variables

This material tracks three history variables:

1. J or the volumetric strain (i.e., the determinant of the deformation gradient, but when initial pressure is set, it will be the product of the determinant of the tracked deformation gradient and the volume change caused by the initial pressure).
2. Jres or the volume change that would occur due to free thermal expansion.
3. The shear rate seen by the particle, which is equal to [math]\displaystyle{ 2|{\rm dev}\nabla(\vec v)| }[/math]

Notes

More precise empirical fits of experimental data to the Tait equation often allows bulk modulus and thermal expansion coefficient to depend on temperature. A common fitting procedure is to define:

      [math]\displaystyle{ B(T) = B_0 e^{-B_1T} }[/math]

      [math]\displaystyle{ V(0,T) = A_0 + A_1T + A_2T^2 + \cdots }[/math]

where B_i and A_i are fitting parameters, which are tabulated for many liquids and even for amorphous polymers. If needed, these refinements may be added in the future.

Examples

The following commands are for water.

<Material Type="27" Name="Water">
  <K>2200</K>
  <viscosity>1</viscosity>
  <alpha>70</alpha>
  <rho>1</rho>
  <Cv>418.13</Cv>
  <kCond>0.58</kCond>
</Material>