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14.5: C.5 4th-order isotropic tensors

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  • To identify the isotropic 4th-order tensors, one uses the same logic as in the 3rd-order case (section C.4) but, as you might guess, there is considerably more of it. The details may be found, for example, in Aris (1962). Here we will just quote the result. The most general isotropic 4th-order tensor is a bilinear combination of 2nd-order isotropic tensors:

    \[A_{i j k l}=\lambda \delta_{i j} \delta_{k l}+\mu \delta_{i k} \delta_{j l}+\gamma \delta_{i l} \delta_{j k},\label{eqn:1}\]

    where \(\lambda\), \(\mu\) and \(\gamma\) are scalars.