What does elasticity with torque and twist depending on normal strain and normal stress look like?

Question

A twisted rope has a relation between normal stress, along the rope, and twist of the rope. If a rope is twisted more it becomes shorter. If it is untwisted it becomes longer. Likewise for coils and helical structures, like DNA and other molecular helices like those common in enzymes. Their elasticity is not described by the reduced elasticity tensor with 21 components. If the constraint of no twisting strain or no torque is avoided in elasticity what does it look like?

I mean an effect like in this animation where compression and elongation of a coiled spring make the ends turn.

Wikipedia bluntly rejects such relations and just says there are symmetries in the elasticity tensor:

The equation for Hooke's law is: ${\displaystyle \sigma _{ij}=C_{ijkl}\,\varepsilon _{kl}\,\!} $ , where $C_{ijkl}$ is the stiffness tensor. These are 6 independent equations relating stresses and strains. The requirement of the symmetry of the stress and strain tensors lead to equality of many of the elastic constants, reducing the number of different elements to 21 $C_{ijkl}=C_{klij}=C_{jikl}=C_{ijlk}$.

Answer 1

Y. C. Fung said "It should be noted that if an external moment proportional to the volume does exist, then the symmetry condition (of a Cauchy stress tensor) does not hold" (p. 68),
and "the vanishing of the symmetric strain tensor $E_{ij}$ or $e_{ij}$ is the necessary and sufficient condition for a neighborhood of a particle to be moved like a rigid body" (p. 96), in "Foundations of Solid Mechanics".
However, I think that a stable elastic body has no rigid-body deformation
and requires a theory other than the symmetric stress and strain tensors.