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 \left(\nabla \mathbf{v} + \nabla \mathbf{v}^T - {2\over 3}{\rm Tr}(\nabla \mathbf{v})\mathbf{I}\right) }[/math]

where [math]\displaystyle{ \nabla \mathbf{v} }[/math] is the velocity gradient and [math]\displaystyle{ \eta }[/math] is the viscosity. The total stress is given by [math]\displaystyle{ \mathbf{\sigma} = -p \mathbf{I} + \tau }[/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	MPa	none
viscosity	Liquid viscosity	cP	none
alpha	Linear thermal expansion coefficient (β0 = 3α)	ppm/K	0
(other)	Properties common to all materials	varies	varies

History Variables

This material uses history #1 to store the volumetric strain (i.e., the determinant of the deformation gradient).

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>