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2.4.2 Isotropic Elasticity in Principal Directions
For isotropic materials, the principal directions of the strain and stress tensors are the same. Let the eigenvalues of be denoted by ( , then the strain energy density may be given as a function of these eigenvalues, . To derive the expression for the stress, recognize that where the are the eigenvectors of . It follows that the second Piola-Kirchhoff stress may be represented as where To evaluate the material elasticity tensor, recognize that where form a permutation over . Then it can be shown that the material elasticity tensor is given by When eigenvalues coincide, L'Hospital's rule may be used to evalue the coefficient in the last term, The double summations in (2.4.2-5) are arranged such that the summands represent fourth-order tensors with major and minor symmetries.
In the spatial frame, the Cauchy stress is given by where and are the eigenvectors of . The principal normal stresses are The spatial elasticity tensor is given by