Tait Liquid Material

Constitutive Law

This MPM material 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) = { K(0,T)\over C } }[/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 β(0) 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 been assumed to be independent of temperature. This material is equivalent to a hyperelastic material with volumetric strain energy function of

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

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

This material is assume to be a Newtonian fluid, which means that the shear stress is given by

      [math]\displaystyle{ \tau = \gamma \nabla \mathbf{v} }[/math]

where [math]\displaystyle{ \nabla \mathbf{v} }[/math] is the velocity gradient. The total stress is given by [math]\displaystyle{ \mathbf{\sigma} = -p \mathbf{I} + \tau }[/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 (β(0) = 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).

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>