Isotropic, Hyperelastic-Plastic Material

Constitutive Law

The Mooney-Rivlin material Mooney-Rivlin is an isotropic, elastic material in large strains using the hyperelastic formulation. The constitutive law derived from a strain energy function that is a function of the deformation gradient tensor F.

Regarding to the objectivity conditions and using the representation theorem, the strain energy function is a function of the invariants of a strain tensor, such as the left Cauchy-Green strain tensor. In the following, it is represented by its representation to volumetric/deviatoric expression:

      [math]\displaystyle{ W =U(J) + {G_{1} \over 2 } (\bar I_{1}-3) + {G_{2} \over 2 }(\bar I_{2}-3) }[/math]

with 3 possible forms of the volumetric energy term U(J):

[math]\displaystyle{ U(J) ={K\over 2 }(J-1)^2 {\rm ,} \qquad U(J) ={K\over 2 }{1\over 2 }((J^2-1)-ln J) \qquad {\rm and} \qquad U(J) ={K\over 2 }(ln J)^2 }[/math]

where [math]\displaystyle{ G_{1} }[/math], [math]\displaystyle{ G_{2} }[/math] and K are material properties, [math]\displaystyle{ \bar I_{1} }[/math], [math]\displaystyle{ \bar I_{2} }[/math] and J are the invariant of the chosen strain tensor, with J=det F and

[math]\displaystyle{ \bar I_{1} ={B_{xx}+B_{yy}+B_{zz} \over J^{2/3}} \qquad {\rm and} \qquad \bar I_{2} ={1 \over 2} (\bar I_{1}^2-{{B_{xx}^2+B_{yy}^2+B_{zz}^2+2B_{xy}^2+2B_{xz}^2+2B_{yz}^2} \over J^{2/3}}) }[/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>