Isotropic, Hyperelastic-Plastic Material

Constitutive Law

This MPM Material is an isotropic, elastic-plastic material in large strains using a hyperelastic formulation. The elastic regime for this material is identical to a Mooney except that it only allows a Neohookean elastic regime (with G = G₁ and G₂ = 0).,

The formulation of finite strain plasticity is based on the notion of a stress free intermediate configuration and uses a multiplicative decomposition of the deformation gradient F given by:

      [math]\displaystyle{ \mathbf{F} = \mathbf{F}_{e}. \mathbf{F}_{p} }[/math]

Where F_e and F_p are the elastic and plastic deformation gradient tensors respectively, with det (F_p), that supposes the plastic flow to be isochoric. The Neo-Hookean elastic stored energy, represented by its uncoupled volumetric-deviatoric internal energy form, is consistent with the fundamental idea that the elastic-plastic deviatoric response is assumed to be uncoupled from the elastic volumetric response

In finite strain plasticity, the stored energy is based on the additive decomposition of the stored energy into elastic W_e and plastic F_p internal energies. The elastic stored energy is related to the intermediate configuration and the plastic stored energy is expressed in term of plastic state variables α.

      [math]\displaystyle{ W =W_{e} (\mathbf{B}_{e}) + W_{p} (\alpha) }[/math]

The stored Neo-Hookean stored energy, W_e, is identical to the Mooney Energy W and dependent on entered small-strain, bulk modulus (κ), small-strain, shear modulus (G = G₁), and dilation energy option (UJOption). The value of G₂ is always zero in this material.

In associative plasticity, the plastic storage energy is represented by the plastic flow condition. The plastic flow model considered here is isotropic hardening. It is handled by any hardening law available in the code (see Hardening Laws). The associative flow rate is defined by the principle of maximum plastic dissipation. It is given, in the present context, by:

      [math]\displaystyle{ L_{v} \mathbf{B}^e = \mathbf{F} {\delta\over t} (\mathbf{\bar C}{^{p-1}}) \mathbf{F^T} = - {2\over 3} {\gamma} {\rm Tr}(\mathbf{B}_{e})\mathbf{n} \qquad {\rm with} \qquad \mathbf{n} = {\mathbf{\tau^{d}}\over ||\mathbf{\tau^{d}}||} }[/math]

Where L_v is the Lie derivative of the deviatoric part of the elastic left Cauchy-Green strain tensor [math]\displaystyle{ \bigl(\mathbf{\bar B}_{e}\bigr) }[/math]. It represents the plastic strain rate that is a tensor normal to the yield surface in the stress space; n is a normal to the yield surface and γ is the consistency parameter also called the plastic multiplicator. In addition, a isotropic hardening law is needed. It is represented by the rate equation, as in the linear theory:

      [math]\displaystyle{ {d {\alpha}\over dt } ={\gamma} \sqrt{2\over3} }[/math]

Material Properties

The material properties are set using

Property	Description	Units	Default
K	Low-strain bulk modulus	pressure units	none
G1 (or G)	Low-strain shear modulus	pressure units	none
UJOption	Set to 0, 1, or 2, to select the energy term from Mooney Material.	none	0
alpha	Thermal expansion coefficient	ppm/K	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)	Properties common to all materials	varies	varies

See these relations to covert other properties (such as modulus and Poisson's ratio) to bulk and shear moduli.

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 elastic left Cauchy Green tensor.

Examples

These commands model a polymer as an isotropic hyperelastic-plastic material with a particular linear isotropic hardening:

Material "polymer","polymer","HEIsotropic"
   K 5000
   G1 1100
   alpha 60
   rho 1.2
   Hardening "Linear"
   yield 72
   Ep 1000   
 Done