View source for Comparison of Neo-Hookean Materials

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== Introduction ==

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

# [[Neo-Hookean Material]]
# [[Mooney Material]]
#* [[Isotropic, Hyperelastic-Plastic Material]]
#* [[Isotropic, Hyperelastic-Plastic Mie-Gr&#252;neisen Material]]
# [[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 (G<sub>1</sub> and G<sub>2</sub>). It can be considered as a neo-Hookeen material that reduces to small strain material with same bulk modulus and shear modulus G = G<sub>1</sub>+ G<sub>2</sub>. The subclasses of [[Mooney Material]] use elastic part of that material with G=G<sub>1</sub> abd G<sub>2</sub>=0. The [[Isotropic, Hyperelastic-Plastic Mie-Gr&#252;neisen Material]] has different elaastic behavior in compression (but not plotted here yet). The third allows plastic deformation by limiting the amount of elongation in tension or compression. By turning off the clamping, the [[Clamped Neo-Hookean Material|clamped neo-Hookean material]] represents a modified corotated elasticity law.

== Elastic Energy Functions ==

The elastic energy functions for these three material compared are:

<ul>
<li>[[Neo-Hookean Material]]<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>W ={\lambda\over 2 }({1\over 2 }(J^2-1)-\ln J) + {G\over 2 } (I_{1}-3-2\ln J) </math><br>
The first term can be changed to two other options to provide more neo-Hookean material models.
</li>
<li>[[Mooney Material]]<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>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><br>
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 G<sub>2</sub>=0 would make it a neo-Hookean material. If all parameters are used, it is neo-Hookean extension of small strain material with G = G<sub>1</sub>+G<sub>2</sub>.
</li>
<li>[[Clamped Neo-Hookean Material]]<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>W = {\lambda\over 2}(J_E-1)^2 + G\sum_k (\lambda_k-1)^2</math><br>
To use this neo-Hookean material, the material properties must set either <tt> CritComp</tt> or <tt> CritTens</tt> to less than zero and set <tt>Elastic</tt> to zero.
</li>
</ul>

== Stress Strain Response ==

The following figure plots stress vs. strain from compression to tension for the above neo-Hookean materials. For the [[Mooney Material]], one plot uses G<sub>2</sub>=0 while the second uses G<sub>1</sub>=G<sub>2</sub>. Although all are the same in the small strain limit the differences at large strain are significant. The [[Mooney Material]] softens at high strain while the [[Neo-Hookean Material]] and [[Clamped Neo-Hookean Material]] (or "Co-Rotated" material) get stiffer at higher strain. The choice of material model should be based on corresponding response of an actual material.

[[File:NewHookean.png|center|600px]]