Active Viscoelasticity of Odd Materials

The mechanical response of active media ranging from biological gels to living tissues is governed by a subtle interplay between viscosity and elasticity. We generalize the canonical Kelvin-Voigt and Maxwell models to active viscoelastic media that break both parity and time-reversal symmetries. The resulting continuum theories exhibit viscous and elastic tensors that are both antisymmetric, or odd, under exchange of pairs of indices. We analyze how these parity violating viscoelastic coefficients determine the relaxation mechanisms and wave-propagation properties of odd materials.

Figure 1

(a) Sketch of the odd Kelvin-Voigt model. Upon application of a constant stress the resulting deformation will be perpendicular to the stress direction. The decay of the deformation will be captured by odd transport coefficients. (b) Depiction of the odd response in odd Kelvin-Voigt materials. It is modeled by a transverse spring and a damper connected in parallel.

Figure 2

(a) Sketch of the odd Maxwell model. A finite deformation will result in a transverse stress. Its relaxation will be captured by odd transport coefficients. (b) Depiction of the odd response in odd Maxwell materials. It is modeled by a spring and a damper connected in series. We emphasize that this collective representation does not imply that odd Maxwell fluids can be constructed out of springs and dampers.

Figure 3

Real and imaginary parts of the frequency eigenvalues. (a),(b) Real and imaginary parts, respectively, of the first eigenvalue. (c),(d) Real and imaginary parts, respectively, of the second eigenvalue. These are obtained for the parameters $k=1$, $\lambda =0$, and ${\rho }_0=1/2$.