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 Jres 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/Jres 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 | voscosity 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 |
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 uses history #1 to store 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).
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 Bi and Ai 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>