Difference between revisions of "Isotropic, Elastic-Plastic Mie-Grüneisen Material"
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<math>{p\over \rho_0} = {C_0^2 \eta \left(1 - {1\over 2}\gamma_0 \eta\right) \over (1 - S_1\eta - S_2\eta^2 - S_3 \eta^3)^2} + \gamma_0 U</math> | <math>{p\over \rho_0} = {C_0^2 \eta \left(1 - {1\over 2}\gamma_0 \eta\right) \over (1 - S_1\eta - S_2\eta^2 - S_3 \eta^3)^2} + \gamma_0 U</math> | ||
where <math>\eta</math> is fraction compression and given by | |||
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<math>\eta = 1 - {\rho_0\over \rho} = 1 - {V\over V_0} = 1 - J</math> | |||
and <math>\gamma_0</math>, <math>C_0</math>, and <math>S_i</math> are material properties and <math>U</math> is total internal energy. The above equation applies only in compression (<math>\eta>0</math>). In tension, the pressure is given by | |||
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<math>P = C_0^2\eta + \gamma_0 U</math> | |||
== Material Properties == | == Material Properties == | ||
Revision as of 11:46, 31 December 2013
Constitutive Law
This MPM material uses a Mie-Grüneisen equation of state in the in the elastic regime and can plastically deform according to any selected hardening law.
Mie-Grüneisen Equation of State
The Mie-Grüneisen equation of state defines the pressure only and the Kirchoff pressure is
[math]\displaystyle{ {p\over \rho_0} = {C_0^2 \eta \left(1 - {1\over 2}\gamma_0 \eta\right) \over (1 - S_1\eta - S_2\eta^2 - S_3 \eta^3)^2} + \gamma_0 U }[/math]
where [math]\displaystyle{ \eta }[/math] is fraction compression and given by
[math]\displaystyle{ \eta = 1 - {\rho_0\over \rho} = 1 - {V\over V_0} = 1 - J }[/math]
and [math]\displaystyle{ \gamma_0 }[/math], [math]\displaystyle{ C_0 }[/math], and [math]\displaystyle{ S_i }[/math] are material properties and [math]\displaystyle{ U }[/math] is total internal energy. The above equation applies only in compression ([math]\displaystyle{ \eta\gt 0 }[/math]). In tension, the pressure is given by
[math]\displaystyle{ P = C_0^2\eta + \gamma_0 U }[/math]
Material Properties
| Property | Description | Units | Default |
|---|---|---|---|
| E | Tensile modulus | MPa | none |
History Variables
None