# 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. See Comparison of Neo-Hookean Materials for details on available neo-Hookean materials.

In this neo-Hookean material, the stored stain energy is given by the expression[1]:

$\displaystyle{ W =U(J) + {G\over 2 } (I_{1}-3-2\ln J) }$

where J (= det F) is relative volume change, G is low strain shear modulus, and I1 is the strain invariant

$\displaystyle{ I_{1} = B_{xx}+B_{yy}+B_{zz} }$

where B is the left Cauchy-Green strain tensor. This material allows three options for the U(J) term:

$\displaystyle{ U(J) ={\lambda\over 2 }({1\over 2 }(J^2-1)-\ln J) \qquad({\rm UJOption}=0) }$

$\displaystyle{ U(J) ={\lambda\over 2 }(J-1)^2 \qquad\qquad\qquad({\rm UJOption}=1) }$

$\displaystyle{ U(J) ={\lambda\over 2 }(\ln J)^2 \qquad\qquad\qquad({\rm UJOption}=2) }$

where $\displaystyle{ \lambda }$ is Lame modulus. At low strains, this material is equivalent to a linear elastic, isotropic material with shear modulus G and bulk modulus $\displaystyle{ \kappa = \lambda+2G/3 }$.

The Cauchy stress tensor is determined by differentiating the strain energy function:

$\displaystyle{ \mathbf{\sigma} ={\lambda\over 2}\left(J-{1\over J}\right)\mathbf{I} + {G \over J } (\mathbf{B}-\mathbf{I}) }$

The above stress is for UJOption=0; for the other two options, the first term changes to:

$\displaystyle{ \lambda(J-1) \quad({\rm UJOption}=1) \qquad{\rm and}\qquad \lambda {\ln J\over J} \quad({\rm UJOption}=2) }$

Note that UJOption=0 is the default option because it is the only one where pressure correctly becomes infinite for both J approaching 0 and J approaching infinity. When using this material in 2D plane stress calculations, the code has to solve for the z direction deformation required to get zero stress in that direction. For UJOption = 0 or 1, the z direction deformation can be found analytically; for UJOption = 2, however, the z direction deformation requires a numerical solution and therefore is less effficient. Note that the neo-Hookean option within the Mooney-Rivilin material needs numerical solutions in plane stress modeling for all UJOptions. Thus, the most efficient neo-Hookean material for plane stress modeling is this material with UJOption = 0 or 1.

The stresses can alternatively be divided into pressure and deviatoric stress, $\displaystyle{ \symbf{\sigma} = -p \mathbf{I} + \bar{\symbf{\sigma}} }$ resulting in:

$\displaystyle{ p = {\lambda\over 2}\left(J-{1\over J}\right) + {G\over J}\left({I_1\over 3}-1\right) }$

$\displaystyle{ \bar{\symbf{\sigma}} = {G \over J }{\rm dev}(\mathbf{B}) }$

The first term in the pressure changes for the other two UJOption settings. Note that the pressure depends on both terms in the strain energy function. This dependence contrasts with the Mooney-Rivilin material where the pressure depends only on the U(J) term.

### 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:

$\displaystyle{ p = {\lambda\over 2}\left(J_{eff}-{1\over J_{eff}}\right) + {G\over J_{eff}}\left({I_1\over 3J_{res}^{2/3}}-1\right) }$

$\displaystyle{ \bar{\symbf{\sigma}} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B}) }$

## Material Properties

The material properties are given in the following table. From the first five, you must enter K and G, Lame and G, OR E and nu. If you enter any other combination or more than two, an error will result.

Property Description Units Default
K The bulk modulus pressure units none
G The shear modulus pressure units none
Lame The Lame modulus pressure units none
E The tensile modulus pressure units 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/K 0
(other) Properties common to all materials varies varies

The first five properties are the resulting low strain properties of this material. These various mechanical properties are interrelated by these equations.

## 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.

## Examples

 Material "matID","My Neohookean Material","Neohookean"
K 1000
G 375
alpha 40
rho 1
UJOption 0
Done


## References

1. R. W. Ogden, "Non-Linear Elastic Deformations," Dover Publications, New York, 1984 (page 222)