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5.2.4 Natural Neo-Hookean
This is a compressible isotropic neo-Hookean material that uses the natural (Hencky) strain tensor invariants to formulate its strain energy density. These invariants are reviewed in . The left Hencky strain is evaluated from where is the left stretch tensor in the polar decomposition of the deformation gradient . To evaluate we first evaluate the left Cauchy-Green tensor from as in eq.(2.3.2-2) and get its eigenvalues and eigenvectors . Then The invariants of the natural strain tensor are where as usual, and It can be shown that Note that as . It also follows that . As explained in , with positive implying expansion and negative implying contraction. Similarly, , with implying no distortion. Finally, with representing uniaxial extension, representing uniaxial contraction and representing pure shear.
For the natural neo-Hookean material the strain energy density is where is the material's bulk modulus and is its shear modulus. To evaluate the Cauchy stress and spatial elasticity tensor , we use the framework of isotropic elasticity in principal directions (Section 2.4.2↑). This requires us to express and in terms of the eigenvalues , Now the stress is given by eq.(2.4.2-7) where, based on eq.(2.4.2-9), the principal normal stresses are evaluated as with forming a permutation over . Similarly, the spatial elasticity tensor is given by eq.(2.4.2-10) where we substitute and Finally, in the limiting case when pairs of eigenvalues are repeated, we substitute