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 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]

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 that is converted to heat.

Material Properties

Property	Description	Units	Default
E	Tensile modulus	MPa	none

History Variables

None