Small-amplitude oscillatory shear (SAOS)

Small-amplitude oscillatory shear probes linear viscoelastic response by imposing $\gamma ={\gamma }_0\sin \omega t$ and resolving the stress into components in phase with strain and strain rate.

The storage modulus is the in-phase coefficient $G^{\prime }$ in $\tau ={\gamma }_0(G^{\prime }\sin \omega t+G^{\prime \prime }\cos \omega t)$. It measures elastic energy storage.

The loss modulus is the quadrature coefficient $G^{\prime \prime }$ in oscillatory shear. It measures viscous energy dissipation.

The loss tangent $\tan \delta =G^{\prime \prime }/G^{\prime }$ compares dissipative and elastic response. Its crossover through one often identifies a dominant viscoelastic relaxation time.

A viscoelastic relaxation time is the characteristic time over which stored stress decays after deformation. A single-mode material commonly crosses from viscous to elastic response around $\omega \lambda =1$.