Neo-Hookean Material
Constitutive Law
This MPM Material is an isotropic, elastic material in large strains using a hyperelastic formulation. It is a neo-Hookean material. Although a Mooney-Rivilin material is a neo-Hookean material when G2=0, this material gives an alternatre neo-Hookean formation with slightly different stresses. This material is only available in OSParticulas.
In this neo-Hookean material, the stored stain energy is given by the expression:
[math]\displaystyle{ W =U(J) + {G\over 2 } (I_{1}-3-2\ln J) }[/math]
where J (= det F) is relative volume change, G is low strain shear modulus, and I1 is the strain invariant
[math]\displaystyle{ I_{1} = B_{xx}+B_{yy}+B_{zz} }[/math]
where B is the left Cauchy-Green strain tensor. This material allows three options for the U(J) term:
[math]\displaystyle{ U(J) ={\lambda\over 2 }({1\over 2 }(J^2-1)-\ln J) \qquad({\rm UJOption}=0) }[/math]
[math]\displaystyle{ U(J) ={\lambda\over 2 }(J-1)^2 \qquad\qquad\qquad({\rm UJOption}=1) }[/math]
[math]\displaystyle{ U(J) ={\lambda\over 2 }(\ln J)^2 \qquad\qquad\qquad({\rm UJOption}=2) }[/math]
where [math]\displaystyle{ \lambda }[/math] is Lame modulus. At low strains, this material is equivalent to a linear elastic, isotropic material with shear modulus G and bulk modulus [math]\displaystyle{ \kappa = \lambda+2G/3 }[/math].
The Cauchy (or true stress) stress tensor is determined by differentiating the strain energy function:
[math]\displaystyle{ \mathbf{\sigma} ={\lambda\over 2}\left(J-{1\over J}\right)\mathbf{I} + {G \over J } (\mathbf{B}-\mathbf{I}) }[/math]
The above stress is for UJOption=0; for the other two options, the first term changes to:
[math]\displaystyle{ \lambda(J-1) \quad({\rm UJOption}=1) \qquad{\rm and}\qquad \lambda {\ln J\over J} \quad({\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.
The stresses can alternatively be divided into pressure and deviatoric stress, [math]\displaystyle{ \mathbf{\sigma} = -p \mathbf{I} + \bar \mathbf{\sigma} }[/math] resulting in:
[math]\displaystyle{ p = {\lambda\over 2}\left(J-{1\over J}\right) + {G\over J}\left({I_1\over 3}-1\right) }[/math]
[math]\displaystyle{ \bar\mathbf{\sigma} = {G \over J }{\rm dev}(\mathbf{B}) }[/math]
The first term in the pressure changes for the other two UJOption settings.
Residual Stress
In the presence of temperature or concentration changes, this material accounts for residual stresses by the process described for the Mooney-Rivlin material The results are:
[math]\displaystyle{ p = {\lambda\over 2}\left(J_{eff}-{1\over J_{eff}}\right) + {G\over J_{eff}}\left({I_1\over 3J_{res}^{2/3}}-1\right) }[/math]
[math]\displaystyle{ \bar\mathbf{\sigma} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B}) }[/math]
Material Properties
The material properties are entered using:
| Property | Description | Units | Default |
|---|---|---|---|
| K | Bulk modulus | MPa | none |
| G | The shear modulus | MPa | none |
| Lame | The Lame modulus | MPa | none |
| E | The tensile modulus | MPa | none |
| nu | The Poisson's ratio | none | none |
| UJOption | Set to 0, 1, or 2, to select the energy term from above. | none | 0 |
| alpha | Thermal expansion coefficient | ppm/M | 0 |
| (other) | Properties common to all materials | varies | varies |
You must enter K and G, Lame and G, OR E and nu. If you enter any other combination of the first five properties, an error will result.
History Data
This material uses history #1 to store the volumetric strain (i.e., the determinant of the deformation gradient or J). The total strain, which is elastic, is stored in the elastic strain variable, while the plastic strain stores the left Cauchy Green strain.