# physical meaning of major symmetry of the stiffness tensor

What happens if a stiffness tensor does not have the "major symmetry" $C_{ijkl}=C_{klij}$?

Background: In linear elasticity (generalising Hooke's law from a spring to a continuous medium), the stiffness tensor calculates the stress (forces) from the strain (deformations), $\sigma_{ij} = C_{ijkl} \varepsilon_{kl}$.

The stiffness tensor must be symmetric in its [$ij$] indices (because stress is, at least in equilibrium), and might as well also be symmetric in its [$kl$] indices (because strain certainly is); these are termed the minor symmetries. Many texts only motivate the major symmetry with half a passing comment on potential energy uniqueness, or the second derivatives thereof. (e.g. Rock physics handbook) This seems unsatisfyingly opaque, particularly since it ought be possible to derive the same physics using forces without reference to energy. Moreover, outside of linear elasticity there seem to be stiffness tensors which do not share the major symmetry (e.g. Inelastic analysis of structures; or Ragione et al 2015 [10.1098/rspa.2015.0013]).

Physically, what does the major symmetry mean, and in precisely what way could the behaviour of a material differ if this was violated?