A neo-Hookean Material

In Chapter 4, we discussed Hooke s law (Eq. (4.6)) for a small strain in one dimension. As long as the applied strain is very small, Eq. (4.6) is valid for a solid material. Here, we generalize it in three dimensions. It appears logical that the stress tensor is linearly proportional to the deformation tensor, that is. 

Here the sign is chosen by taking compression to be positive. For an incompressible material, the total stress tensor is given by 

The mechanical property represented by Eq. (5.50) is often called neo-Hookean. 

We apply Eq. (5.50) to the uniaxial-extension case with 8i being the direction of extension. From Eqs. (5.45) and (5.50), we have 

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Minimization of the total elastic strain energy suggests that the rod will become unstable at a critical amount of torsion part of the rod will unwind and form a tight ring while the remainder of the rod will become slightly more stretched. A simple criterion can be derived on this basis for the onset of kinks. For a neo-Hookean material, Eq. (1.8), the condition for forming a kink becomes ... 

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