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 Mooney_Rivlin.

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.

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]

In low strains, this material is equivalent to a linear elastic, isotropic material with shear modulus G = G₁ + G₂ and bulk modulus K. If G₂ = 0, the material is a neo-Hookean material. See below for an alternate compressibility terms. Some hyperelastic rubber models assume incompressible materials, which corresponds to K → ∞; such models do not work in dynamic code (because wave speed is infinite).  The Cauchy (or true stress) stress tensor is determined by differentiating the strain energy function. It is represented here by the addition of the spheric (pressure) and the deviatoric stress tensors, [math]\displaystyle{ \mathbf{\sigma} = p \mathbf{I} + \bar \mathbf{\sigma} }[/math] given by:

[math]\displaystyle{ \mathbf{\sigma} =K(J-1)\mathbf{I} + {G_{1} \over J^{5/3} } (\mathbf{B}-{I_{1} \over3}\mathbf{I}) + {G_{2} \over J^{7/3}} (I_{1} \mathbf{B}-\mathbf{B^2}-{2I_{2} \over3}\mathbf{I}) }[/math]

where [math]\displaystyle{ I_{1} = J^{2/3} \bar I_{1} \qquad {\rm and} \qquad I_{2} = J^{4/3} \bar I_{2} }[/math] .

Material Properties

The constants involved in the strain energy function, are equivalent in small strains to properties of isotropic elastic material with Poisson's ratio [math]\displaystyle{ {\nu} }[/math] as well as shear G = G1 + G2 and bulk modulus K given by

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

Property	Description	Units	Default
E	Elastic modulus	MPa	none
G1, G2	Shear modulus	MPa	none
alpha	Thermal expansion coefficient	ppm/M	40

History Variables

None

Examples

These commands model polymer as an isotropic, hyperelastic material with G1=G2 =G/2 (using scripted or XML commands):

Material "polymer","polymer","Mooney"
   E 2500
   nu .4
   alpha 60
   rho 1.2
 Done

 
<Material Type="8" Name="polymer">
   <rho>1.2</rho>
   <G1>35.714285714</G1>
   <G2>35.714285714</G2>
   <K>166.66666666</K>
   <alpha>60</alpha>
 </Material>