Difference between revisions of "Mooney Material"
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<math>W = | <math>W =U(J) + {G_{1} \over 2 } (\bar I_{1}-3) + {G_{2} \over 2 }(\bar I_{2}-3)</math> | ||
with | |||
<math> U(J) ={K\over 2 }(J-1)^2 </math> | |||
<math> U(J) ={K\over 2 }(J-1)^2 </math> | |||
<math> U(J) ={K\over 2 }(J-1)^2 </math> | |||
Revision as of 17:11, 21 September 2013
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 [math]\displaystyle{ U(J) ={K\over 2 }(J-1)^2 }[/math]
[math]\displaystyle{ U(J) ={K\over 2 }(J-1)^2 }[/math]
[math]\displaystyle{ U(J) ={K\over 2 }(J-1)^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}} }[/math]
[math]\displaystyle{ \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 shear modulus 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>