View source for Tait Liquid Material

== Constitutive Law ==

This [[Material Models|MPM material]] models a liquid as a hyperelastic material. The pressure in the liquid is found from the Tait equation:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>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

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>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<sub>res</sub>'' as determinant of deformation gradient due to free thermal expansion, or:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>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 ''T''<sub>0</sub> is the [[Thermal Calculations#Stress Free Temperature|stress free temperature]] and &beta;(0) is the zero-pressure, volumetric, thermal expansion coefficient (which has been assumed to be independent of temperature), the constitutive law for pressure is:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>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 ''K''<sub>0</sub>, and it has also been assumed to be independent of temperature (''i.e''., ''B''(''T'') = ''K''<sub>0</sub>/''C''). This material is equivalent to a hyperelastic material with volumetric strain energy function of
      
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>U(J^*) = C K_0\left[ C \exp\left({1-J^*\over C}\right) + J^*\right]</math>

where ''J''<sup>*</sup> = ''J''/''J<sub>res</sub>'' is the effective volumetric ratio. This energy function is equivalent to the energy per unit initial volume for isothermal compression or expansion of a Tait liquid.

For shear stress calculations, this material is assumed to be a Newtonian fluid, which means that the shear stress is given by

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\tau = \gamma \nabla \mathbf{v}</math>

where <math>\nabla \mathbf{v}</math> is the velocity gradient. The total stress is given by <math> \mathbf{\sigma} = -p \mathbf{I}  + \tau</math>

=== Pressure-Dependent Properties ===

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

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>K(P,T) = {p + B(T)\over C} {J\over J_{res}}</math>

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

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\beta(P,T) = \beta(0,T) + {P\over K(P,T)B(T)} {dB(T)\over dT}</math>

where

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\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 reduce to:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>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:

{| class="wikitable"
|-
! Property !! Description !! Units !! Default
|-
| K || Zero-pressure, bulk modulus || MPa || none
|-
| viscosity || Liquid viscosity || cP || none
|-
| alpha || Linear thermal expansion coefficient (&beta;(0) = 3&alpha;) || ppm/K || 0
|-
| ([[Common Material Properties|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:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>B(T) = B_0 e^{-B_1T}</math>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>V(0,T) = A_0 + A_1T + A_2T^2 + \cdots</math>

where ''B<sub>i</sub>'' and ''A<sub>i</sub>'' 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>