Comparison of Neo-Hookean Materials

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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:

  • 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.
  • 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]
    The first term can be changed to two other options to provide more neo-Hookean material models.This material is not strictly neo-Hookean (because of three material properties), but choosing G2=0 would make it a neo-Hookean material. If all parameters are used, it is neo-Hookean extension of small strain material with G = G1+G2.
  • Clamped Neo-Hookean Material
           [math]\displaystyle{ W = {\lambda\over 2}(J_E-1)^2 + G\sum_k (\lambda_k-1)^2 }[/math]
    To use this neo-Hookean material, the material properties must set either CritComp or CritTens to less than zero and set Elastic is 0.
    • Stress Strain Response

      The following figure plots stress vs. strain from compression to tension for the above neo-Hookean materials. For the Mooney Material