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?

## Best Answer

As you probably realized, violation would allow perpetual motion. In the simplest example, just set all Cijkl to zero except for Cyyxx, so horizontal compression changes the vertical force, but vertical compression does nothing. Then, just compress vertically, compress horizontally, expand vertically, and expand horizontally, in that order. You get a higher force back from the vertical expansion than you needed for the vertical compression.

"To derive the same physics using forces without reference to energy", you could use a lower-level representation of atoms with equal and opposite pairwise forces, thereby maintaining Newton's third law.

The inelastic structures you mentioned are probably getting the energy for such a cycle from within (and their Cijkl tensor correspondingly changes to become symmetric over time).