Difference between revisions of "Tait Liquid Material"
| Line 9: | Line 9: | ||
| | ||
<math>B(T) = { K(0,T) | <math>B(T) = { K(0,T)\over C }</math> | ||
where K(0,T) is the temperature dependence of the bulk modulus at zero pressure. | where K(0,T) is the temperature dependence of the bulk modulus at zero pressure. Defining ''J'' as relative volume (''i.e.'', determinant for deformation gradient) and ''J<sub>res</sub>'' as determinant of deformation gradient due to free thermal expansion, or: | ||
| |||
<math>J = {V(P,T)\over V(0,T_0)} \qquad {\rm and} \qquad J_{res} = {V(0,T)\over V(0,T_0)}</math> | |||
where T<sub>0</sub> is the [[Thermal Calculations#Stress Free Temperature|stress free temperature]], the constitutive law for pressure is: | |||
| |||
<math>P = CK_0\left[\exp\left({1\over C}\left(1 - {J\over J_{res}}\right)\right)-1\right]</math> | |||
== Material Properties == | == Material Properties == | ||
Revision as of 17:53, 30 December 2013
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 for 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)} }[/math]
where T0 is the stress free 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]
Material Properties
| Property | Description | Units | Default |
|---|
History Variables
None