Comparison of Neo-Hookean Materials

Introduction

The definition of an isotropic, neo-Hookean is an elastic material that depends ot two elastic constants and in the limit of small deformations is identical to a small-strain, linear-elastic, Hookean material. The currently available MPM Materials that a neo-Hookean material options are:

1. Neo-Hookean Material
2. Mooney Material
3. Clamped Neo-Hookean Material

The first is always a neo-Hookean material. The second has three properties for bulk modulus (K) and two shear modulus (G₁ and G₂). Is can be considered as neo-Hooken that reduce to small strain material with same bulk modulus and shear modulus G = G₁+ G₂. The third allows plastic deformation by limiting the amount of elongation in tension or compressure. By turning off the clamping, the clamped neo-Hookean material represents a modified corotated elasticity law.

Elastic Energy Functions

The elastic energy functions for these three material compared are:

1. Neo-Hookean Material
          [math]\displaystyle{ W ={\lambda\over 2 }({1\over 2 }(J^2-1)-\ln J) + {G\over 2 } (I_{1}-3-2\ln J) }[/math]
   The first term can be changed to two other options to provide more neo-Hookean material models.
2. Mooney Material
          [math]\displaystyle{ W ={\kappa\over 2 }({1\over 2 }(J^2-1)-\ln J) + {G_{1} \over 2 } (\bar I_{1}-3) + {G_{2} \over 2 }(\bar I_{2}-3) }[/math]
3. Mooney Material
          [math]\displaystyle{ W = {\lambda\over 2}(J_E-1)^2 + G\sum_k (\lambda_k-1)^2 }[/math]
The first two materials can option change the first term to give more neo-Hookean material models. The Mooney Material is not strictly neo-Hookean (because of three material properties), but choosing G₂=0 would make it a neo-Hookean material.