Mooney Material
Constitutive Law
The Mooney-Rivlin material Mooney-Rivlin is an isotropic, elastic material in large strains using the hyperelastic formulation.
Within the framework of hyperelasticity, the existence of a stored-energy W (per unit deformed or indeformed volume), function of a deformation gradient tensor (F, is postulated (Weichert et al., 2000; Truesdell and Noll 1965 and Ogden, 1984) and the constitutive law derived from a strain energy function that is a function of the deformation gradient tensor F, according to the inequality of Clausius-Duheim. Regarding to the objectivity conditions and using the representation theorem (Truesdell and Noll, 1965), the strain energy function is a function of the invariants of a strain tensor. With the left Cauchy-Green strain tensor, Cauchy stress is given by:
[math]\displaystyle{ \mathbf{\sigma} =2 {\delta W \over {\delta \mathbf{B}}} \mathbf{B} }[/math]
In a Mooney-Rivlin material, stored stain energy is give by the expression:
[math]\displaystyle{ W =U(J) + {G_{1} \over 2 } (\bar I_{1}-3) + {G_{2} \over 2 }(\bar I_{2}-3) }[/math]
where J (= det F) is relative volume change, G1 and G2 are shear material properties, and [math]\displaystyle{ \bar I_{1} }[/math] and [math]\displaystyle{ \bar I_{2} }[/math] are the strain invarients:
[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]
This material allows three options for the volumetric energy term:
[math]\displaystyle{ U(J) ={\kappa\over 2 }({1\over 2 }(J^2-1)-\ln J) \qquad({\rm UJOption}=0) }[/math]
[math]\displaystyle{ U(J) ={\kappa\over 2 }(J-1)^2 \qquad\qquad\qquad({\rm UJOption}=1) }[/math]
[math]\displaystyle{ U(J) ={\kappa\over 2 }(\ln J)^2 \qquad\qquad\qquad({\rm UJOption}=2) }[/math]
where [math]\displaystyle{ \kappa }[/math] is bulk modulus. At low strains, this material is equivalent to a linear elastic, isotropic material with shear modulus G = G1 + G2 and bulk modulus [math]\displaystyle{ \kappa }[/math]. 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 [math]\displaystyle{ \kappa\to\infty }[/math]; such models do not work in dynamic code (because the dilational wave speed is infinite).
The Cauchy (or true stress) stress tensor is determined by differentiating the strain energy function. It is calculated by the addition of press and deviatoric stress, [math]\displaystyle{ \mathbf{\sigma} = -p \mathbf{I} + \bar \mathbf{\sigma} }[/math] resulting in:
[math]\displaystyle{ \mathbf{\sigma} ={\kappa\over 2}\left(J-{1\over J}\right)\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} }[/math] and [math]\displaystyle{ I_{2} = J^{4/3} \bar I_{2} }[/math]. The above stress assumes the default UJOption=0. For the other options, the pressure term changes to
[math]\displaystyle{ p = -\kappa(J-1) \qquad\qquad({\rm UJOption}=1) }[/math]
[math]\displaystyle{ p = -\kappa {\ln J\over J} \qquad\qquad({\rm UJOption}=2) }[/math]
Note that UJOption=0 is the default option because it is the only pressure that correctly becomes infinite for both J approaching 0 and J approaching infinity.
Ideal Rubber
This material provides an acceptable material model for rubbers or elastomers, but it will not get the thermodynamics correct unless you indicate the material is an ideal rubber (using the IdealRubber property). An ideal rubber means that during isothermal loading, there is no change in internal energy. In other words, all the work is turned into heat. In adiabatic loading, this heat causes the temperature to rise. For the thermal calculations to be correct, the material must specify a valid heat capacity.
Material Properties
The material properties are enters using:
| Property | Description | Units | Default |
|---|---|---|---|
| K | Bulk modulus | MPa | none |
| G1 | The G1 shear modulus | MPa | none |
| G2 | The G2 shear modulus | MPa | 0 |
| UJOption | Set to 0, 1, or 2, to select the energy term from above. | none | 0 |
| alpha | Thermal expansion coefficient | ppm/M | 0 |
| kCond | Thermal conductivity | W/(m-K) | 0 |
| beta | Solvent expansion coefficient | 1/(wt fraction) | 0 |
| D | Solvent diffusion constant | mm2/sec | 0 |
| (other) | Properties common to all materials | varies | varies |
History Variables
This material uses history #1 to store the volumetric strain (i.e., the determinant of the deformation gradient). The total strain, which is elastic, is stored in the elastic strain variable, while the plastic strain stores the left Cauchy Green tensor.
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>
References
• Ogden R. W., (1984), Non-linear elastic deformations. Wiley et Sons, New York.
• Truesdell C. and W. Noll W (1965), The nonlienar field theories of mechanics, Edition Handbuch der Physik, Vol. III. Spinger, Berlin.
• Weichert D. et Y. Basar, (2000), Nonlinear continuum mechanics of solids, Springer, New York.