Neo-Hookean Material

Constitutive Law

This MPM Material is an isotropic, elastic material in large strains using a hyperelastic formulation (and currently only available in OSParticulas). 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.

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 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. 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, [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. 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:

      [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 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	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

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.