Isotropic, Elastic-Plastic Mie-Grüneisen Material

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 [math]\displaystyle{ C_0 }[/math] property if the bulk wave speed under low-pressure conditions. It is related to the low pressure bulk modulus by:

      [math]\displaystyle{ K_0 = \rho_0 C_0^2 }[/math]

The above pressure equation is used 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]

This equation of state also causes a temperature change of

      [math]\displaystyle{ dT = -JT \gamma_0 {V(t+\Delta t)-V(t)\over V} + {dq \over C_V} }[/math]

where dq is dissipated energy, such as plastic energy, that is converted to heat. By including temperature rises and internal energy, this material automatically thermally expands with the appropriate thermal expansion coefficient without needing to enter a thermal expansion coefficient. The linear thermal expansion coefficient the results is

      [math]\displaystyle{ \alpha = {\rho_0\gamma_0 C_v\over 3K_0} }[/math]

For more details on the Mie-Grüneisen equation of state, you can refer to Wilkens (1999). That reference also has a table of experimentally determined Mie-Grüneisen for numerous materials.

Material Properties

The Mie-Grüneisen equation of state properties and the hardening law properties are set with the following options:

Property	Description	Units	Default
C0	The bulk wave speed, which is equal to [math]\displaystyle{ \sqrt{K/\rho} }[/math]	m/sec	4004
gamma0	The [math]\displaystyle{ \gamma_0 }[/math] parameter	none	1.64
Hardening	This command selects the hardening law by its name or number. It should be before entering any yielding properties.	none	none
(yield)	Enter all plasticity properties required by the selected hardening law.	varies	varies
(other)	All other properties are identical to the properties for an isotropic material.	varies	varies

History Variables

None