Minor and major symmetries of linear elasticity ($C_{ijkl}=C_{jikl}=C_{ijlk},C_{ijkl}=C_{klij}$)

A linear map from symmetric three-dimensional infinitesimal strain tensors to symmetric stress tensors acts between two six-dimensional spaces. Choosing the minor index symmetries $C_{ijkl}=C_{jikl}=C_{ijlk}$ leaves $6\times 6=36$ independent constitutive coefficients. Symmetry of the two second-rank tensors alone does not force pair interchange. If a strain-energy density exists and stress is its derivative, equality of mixed second derivatives gives the major symmetry $C_{ijkl}=C_{klij}$. The resulting symmetric six-by-six coefficient matrix has $6\cdot 7/2=21$ independent coefficients. This distinguishes minor tensor symmetry from the extra reciprocity supplied by an energy potential.