# 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?

## 1 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).

• This is exactly analogous to the proof of symmetry of mutual capacitances in Purcell, if I remember correctly, and that's a good place to go to for a comparison. – Emilio Pisanty Dec 7 '15 at 0:33
• So it seems to be a matter of ensuring that the stresses are conservative, hence writing the work done on the material body as a gradient of something. (It would be inconvenient to show purely with forces, because you would need to detail the particles/mechanism of the external agent, to be able to say whether the external universe has to change for the material body to return to the beginning of a cycle through its configuration space..) Violation may correspond to faulting, plasticity, energy dissipation.. – benjimin Dec 12 '15 at 4:10