Isotropic, Hyperelastic-Plastic Material

Constitutive Law

The HEIsotropic material HEIsotropic is an isotropic, elastic-plastic material in large strains using the hyperelastic formulation, using a Neo-Hookean strain energy function (see Mooney-Rivlin) (Simo J C and T J R Hughes, 2000).

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 Fe and Fp 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 We and plastic Wp 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 particular Neo-Hookean stored energy considered here is recalled:       [math]\displaystyle{ W_{e} ={\kappa\over 2 }({1\over 2 }(J_{e}^2-1)-ln J_{e}) + {G \over 2 } (\bar I_{1e}-3) }[/math]

Where J_e=det (F)_e and [math]\displaystyle{ \bar I_{1e} = Trace(\mathbf{\bar B}_{e}) = Trace(\mathbf{\bar F}_{e} \mathbf{\bar F}_{e}^T) }[/math] is the deviatoric part of the left Cauchy-Green strain tensor and [math]\displaystyle{ \bar\mathbf{\bar F}_{e} = J_{e}^{-1/3} \mathbf{\bar F}_{e} }[/math] is the deviatoric part of the elastic deformation gradient. κ and G are interpreted in small strains as bulk and shear modulus respectively.

The elastic stress-strain constitutive law, which derives from the elastic storage energy, is given here in term of Kirchhoff stress tensor by:

[math]\displaystyle{ \mathbf{\tau} = J_{e} p \mathbf{I} + G dev (\bar \mathbf{B}_{e}) }[/math]

dev is the deviatoric part of the considered tensor, and [math]\displaystyle{ p = U'( J_{e}) = {\kappa\over 2} ( {(J_{e}^2-1)\over J_{e}} }[/math]

In the case of associate plasticity, the plastic storage energy is represented by the plastic flow condition. The plastic flow model considered here is the isotropic hardening. It is handled by any hardening law available in the code (see Hardening Laws).

The associate flow rate is defined by the principle of maximum plastic dissipation (Simo J C, 1988a and 1988b). 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} Trace(\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 elastic left Cauchy-Green strain tensor, 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 γ is the consistency parameter also called the plastic multiplicator.

In addition, the isotropic hardening law is needed. It is represented by the rate equation, as in the linear theory:

[math]\displaystyle{ {d {\alpha}\over dt } = ({2\over 3})^{1/2} {\gamma} }[/math]

Material Properties

The constants involved in the strain energy function, are equivalent in small strains to the properties of isotropic elastic material with Poisson's ratio [math]\displaystyle{ {\nu} }[/math] as well as shear modulus G and bulk modulus [math]\displaystyle{ {\kappa} }[/math] given by

[math]\displaystyle{ G = {E \over 2({1+\nu })} \qquad {\rm and} \qquad {\kappa} = {E \over 3({1-2\nu })} }[/math].

with:

Property	Description	Units	Default
E	Elastic modulus	MPa	none
[math]\displaystyle{ {\nu} }[/math]	Poisson's ratio	none	none
alpha	Thermal expansion coefficient	ppm/M	40

History Variables

None

Examples

These commands model polymer as an isotropic hyperelastic-plastic material with a particular linear isotropic hardening (using scripted or XML commands):

Material "polymer","polymer","HEIsotropic"
   E 3100
   nu .4
   yield 72
   Ep 1000   
   alpha 60
   rho 1.2
 Done

 
 <Material Type="24" Name="Polymer">
   <rho>1.2</rho>
   <K>5166.67</K>
   <G1>1107.14</G1>
   <yield>72</yield>
   <Ep>1000</Ep>
   <alpha>60</alpha>
 </Material>

G1 represents G in the formulation on top

References

• Simo J. C. and T. J. R. Hughes (2000), "Computational Inelasticy", Interdisciplinary Applied Mechanics, Volume 7. Springer Edition.

• Simo J. C. (1988a), "Framework for finite elastoplasticity. Part I", Computer Methods in Applied Mechanics and Engineering, 66: 199-219.

• Simo J. C. (1988b), "Framework for finite elastoplasticity based on maximum dissipated energy and the multiplicative decomposition. Part II: Computational aspects", Computer Methods in Applied Mechanics and Engineering, 68: 1-31.