Isotropic, Hyperelastic-Plastic Mie-Grüneisen Material

Constitutive Law

This MPM material is identical to an HEIsotropic material except that it uses a Mie-Grüneisen Equation of State|Mie-Grüneisen equation of state in the elastic regime. The elastic-plastic shear response is handled using the shear terms in the HEIsotropic material. A small-strain version of this material used to be available (with ID=17), but has been deleted. For compatibility, old files that used this material are automatically converted to this hyperelastic material instead.

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 is 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]

Note that the Mie-Grüneisen parameters are typically determined by fitting to experimental or molecular modeling results. But such fits are only valid over the range of [math]\displaystyle{ \eta }[/math] represented by the experiments or modeling. If extended beyond that range (to higher [math]\displaystyle{ \eta }[/math]), the denominator may cross zero or reach a positive minimum and start increasing. The former causes a singularity in bulk modulus followed by negative modulus; the latter causes bulk modulus to decrease with further compression. Both these situations are non-physical can be avoided by using the Kmax material property that defines the maximum increase in bulk modulus. If the effective bulk modulus increases beyond this limit at high [math]\displaystyle{ \eta }[/math], it is set to the maximum value instead.

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 S₃ parameters while Wilkens only has S = S₁). The Wilkens reference also has a table of experimentally determined Mie-Grüneisen properties for numerous materials (although these properties have only S₁ = S for the denominator).

Tension and Shear Response

The above Mie-Grüneisen pressure equation is used only in compression ([math]\displaystyle{ \eta\gt 0 }[/math]). In tension, the pressure is given by the hyperelastic response of

      [math]\displaystyle{ P = -\frac{U(J_{eff})}{dJ_{eff}} }[/math]

where J_eff = J/J_res, where J_res is the relative volume for free thermal expansion and U(J_eff) is one of the three strain energy options defined for a Mooney Material where κ = K₀

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).

Thermal Effects and Thermal Expansion

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 that 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. Also note that any thermal expansion coefficient you enter will be ignored and replaced by the above result.

Moisture Expansion

This material current does not expand or contract due to moisture content. If you enter a moisture expansion coefficient, it will be ignored. This situation may change in the future, but existing derivations of the Mie-Grüneisen equation of state current do not account for moisture related residual strains.

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	alt velocity units	4004
gamma0	The γ0 parameter	none	1.64
S1	The S1 parameter	none	1.35
S2	The S2 parameter	none	0
S3	The S3 parameter	none	0
UJOption	Set to 0, 1, or 2, to select the energy term for tensile loading from Mooney Material	none	0
Kmax	Maximum bulk modulus (relative to zero-pressure bulk modulus) allowed	none	-1
G (or G1)	Low-strain shear modulus	pressure units	none
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)	Properties common to all materials	varies	varies

Note: The first time the relative bulk modulus exceeds Kmax, a warning is printed, modulus is limited, and calculations continue. The default value for Kmax is -1, which means to not limit the bulk modulus. This mode is almost always stable, but simulations with high compression should always add the AdjustTimeStep Custom Task to keep calculation stable under high tangent bulk modulus conditions.

History Variables

The selected hardening law will create one or more history variables. This material uses the next history variable (after the hardening laws history variables) to store the volumetric change (i.e., J or the determinant of the deformation gradient). The total strain is stored in the elastic strain variable, while the plastic strain stores the left Cauchy Green tensor.

Examples

Material "copper","Copper","HEMGEOSMaterial"
  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","HEMGEOSMaterial"
  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