Difference between revisions of "Tait Liquid Material"
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<math>V( | <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 | where C = 0.0894 is a universal Tait constant, V(0,T) is the temperature dependence of the volume at zero pressure, and | ||
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<math>J = {V( | <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 β<sub>0</sub> is the zero-pressure thermal expansion coefficient (which has been assumed to be independent of temperature), the constitutive law for pressure is: | where T<sub>0</sub> is the [[Thermal Calculations#Stress Free Temperature|stress free temperature]] and β<sub>0</sub> is the zero-pressure thermal expansion coefficient (which has been assumed to be independent of temperature), the constitutive law for pressure is: | ||
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<math> | <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 been assumed to be independent of temperature. | Here the zero-pressure bulk modulus is ''K''<sub>0</sub>. and it has been assumed to be independent of temperature. | ||
Revision as of 18:00, 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 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 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.
Material Properties
| Property | Description | Units | Default |
|---|
History Variables
None