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} = {\gamma_0 C_v\over 3C_0^2} }[/math]

Note that thermal expansion depends on C_v, which means you must always enter a valid heat capacity for this material, otherwise the thermal expansion will be wrong.

For more details on the Mie-Grüneisen equation of state, you can refer to Wilkens (1999). The pressure equation here is different than Wilkens, but is equivalent if compared as polynomial expansions; this form is more general because it includes S₂ and S₁ options while Wilkens only has S = S₁). The Wilkens reference also has a table of experimentally determined Mie-Grüneisen for numerous materials (although the fits have only S₁ = S for the denominator).

The shear stress is related to deviatoric strain by the material's shear modulus. The shear modulus is a constant (unless it is changed by a hardening law).

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	m/sec	4004
gamma0	The γ0 parameter	none	1.64
S0	The S0 parameter	none	1.35
S1	The S1 parameter	none	0
S2	The S2 parameter	none	0
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, except that only shear modulus, G, is used and thermal expansion coefficient, alpha, is ignored.	varies	varies

History Variables

The material has none, but the hardening law will have at least one.

In particle properties, the "strain" will be the elastic strain and the "plastic strain" will have the plastic strain. The total strain is the sum of elastic and plastic strains.

Examples

Material "copper","Copper","MGEOSMaterial"
  C0 3933
  S1 1.5
  gamma0 1.99
  rho 8.93
  G 48000
  Cv 134
  kCond 401
  hardening "JohnsonCook"
  Ajc 90
  Bjc 292
  njc .31
  Cjc 0.025
  ep0jc 1
  Tmjc 1356
  mjc 1.09
Done

Material "pmma","PMMA","MGEOSMaterial"
  C0 2300
  S1 1.82
  gamma0 1.82
  rho 1.18
  G 1075
  Cv 1466
  kCond 0.2
  hardening "Linear"
  yield 40
  Ep 1500
Done